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mrs-harkness · 4 years

Text

Run to Me (Part 5)

Pairing: Diane Sherman x Fem!Reader

Wordcount:2.4K

Part 1 || Part 2 || Part 3 || Part 4

Taglist: @fauxplant @deliacoven @escapetodreamworld @kikaykimkim

A/n: Sorry for the delay! I am fighting a serious virus and the holiday... and I may have fallen in love with the real Diane Sherman? I dunno, long story. Anyways, hope you like and Part 6 will be close behind! Love you guys! Thanks for the kind words!

Diane bobbed her head to the music, singing the song on the radio under her breath. You laid your head on the window, the cool glass relieving some of your discomfort. You felt like you were on a roller coaster, the world around you spinning one way then the next. You looked over at her, slightly jealous that she felt fine. She noticed and took her eyes off the road long enough to get a good look at you. She frowned and put her hand on your knee, caressing it with her thumb.

"I can turn around if you want. I think you need the rest," she said, drowning out the music.

You shook your head, groaning. For some reason the pounding behind your eyes grew stronger. You shouldn't have done that.

"I need to see Mandy. I need to talk to her face to face. I won't stay long, but I still need to do this."

Diane bit the corner of her lip, thinking about something. She nodded in response, removing her hand and then sticking it in her coat pocket. She pulled it back out, holding out her hand in front of you. In her palm sat two different pills. You sat up, looking over at her.

"What are those?"

"Aspirin, to help your headache," she said quietly, forcing a smile.

You picked up the pills and looked at them. One was different from the other.

"One of them isn't the same as the other though," you said, wondering if she had accidentally pulled out the wrong pills.

"One is an aspirin, the other is a stronger pain killer. A full dose of aspirin isn't enough to knock that migraine out completely, but a full dose of a pain killer would knock you out, and I know you want to be as coherent as possible during your meeting with Mandy."

Diane smiled nervously at you. You couldn't help but smile back. You knew she didn't want you to go off alone right now, but the fact that she was trying her hardest to make sure you could do what you wanted just made your heart sing. You threw back the pills and swallowed, eager for them to kick in.

Diane watched you for a moment longer and you raised your eyebrows at her, chuckling nervously.

"What? Do I have something on my face?" you asked.

She smiled and blushed, turning her eyes back to the road. Her smile remained though.

"No, just making sure you took your medication since I can't trust you to eat your breakfast," she said, her voice laced with a sassy tone.

You laughed, even though your head protested. You reached over and hit Diane's shoulder and she gasped, holding her arm dramatically.

"Watch it missy, or I'm going to have to punish you."

Suddenly the air in the car seemed thick, and she looked over at you in a way you hadn't seen before. Something was different in her eyes and her smile was curved... differently.

You laughed nervously, unsure of what to do. She looked at you for a moment longer in silence before coming to and turning her eyes back to the road. She cleared her throat and her eyes seemed to dart around the road.

"So I am going to run by the pharmacy when you're with Mandy and pick up some flu medicine. I think that's what going on with you. It wouldn't be surprising after that week you spent cramped on a bus with God knows who," she said. She said it in a completely normal tone as if what just happened, hadn't.

You slumped in your seat and nodded.

"Yeah, sure. Okay," you mumbled as you turned the other way. Diane confused you.

Diane dropped you off a block away from the coffee shop. It was a real fight for her to agree, but she finally said okay if you promised to take the mug of tea you hadn't touched with you. You told her you just wanted to stretch your legs and walk a bit, but really you just didn't want to look like you were a little kid being dropped off by your mom to a playdate. You agreed to meet her back in two hours and you went on your way.

You saw Mandy in the coffee shop window, early as usual. She always liked to be the first one to arrive, no matter what the circumstance. She was on her laptop, sipping her drink, looking beautiful and healthy. You could barely contain your excitement and your chest tightened. You quickly took a puff of your inhaler and ran inside, another sweet little bell signaling your arrival.

Mandy looked up and met your stare, her face nearly splitting in half from her smile. She hopped off the stool and ran to you. You met her in the middle and you crashed into one another's arms. You nuzzled into her hair, letting it hide your tears. You had missed her so much. It had been so long since you had seen her face. She pulled back, looking you over. Immediately her face fell.

"What happened to you? What is this cut? Why are you so pale?" she asked, her hands frantic on your face, turning it from side to side looking for more injures.

You laughed and knocked her hands away.

"Well hello to you too.. I had a little bit of an accident and I think I'm coming down with something, but truly I'm fine."

She lead you to the table and you sat across from her. You forced your face to remain normal, even though climbing onto the stool was painful. You placed your cup on the table and leaned on it. Mandy stared at you for a while.

"So, what was this accident that left a dent in my baby sisters head?"

"Someone accidentally.... hit me with their car?"

The silence between that followed was deafening. Her face was void of any emotion, unblinking.

"I'm sorry, what? Someone did WHAT?" she hissed, suddenly exploding.

You leaned back, afraid Mandy may actually smack you.

"I walked into the street without looking. But it's okay because the person who hit me with their car had medical training! They took care of me. Really, I'm fine."

Mandy looked at you, dumbfounded.

"Okay, we are gonna touch back on that later. Do you want anything to eat or drink? I'll cover you."

You shook your head and pushed the travel mug towards her on the table.

"No I'm okay. I promised I'd drink this."

"Who brings coffee to a coffee shop with them? And what is that? Why does it smell like that?"

"I promised I'd drink it. It's some sort of herbal thing, I dunno. Supposed to help with my immune system or something."

Mandy took a bigger sniff of it, gagging. She pushed it back towards you again.

"Please tell me you didn't take that from someone off the street. You don't know how to say no and Y/n I'm pretty sure that's someone's pis-"

"No! Why don't you have more faith in me? The woman I'm staying with makes it for me," you said, grabbing it and taking sip, suppressing the grimace you felt. You hated it, but she didn't need to know that.

"Ahh okay. Yeah, please tell me more about this mysterious woman who just let a random stranger into her home with no ulterior motives," she said, taking a dramatic sip of her own drink. She coughed, choking on it. You tried to suppress a smile, but you couldn't and she saw it so she kicked you under the table.

"She's not some weirdo. She's actually the one who hit me with her car-"

"OH MY GOD. You're making this up. This can not be real Y/n."

"Would you stop? You make it so difficult to tell a story! Yes, she is the one who hit me with her car. Her name is Diane and-"

"Diane what?"

You looked over at Mandy, angry for her cutting you off once again, but the look on her face made your stomach drop a bit. She wasn't playing. She was serious.

"Uh, Diane? Diane Sherman.." you squeaked, clasping the cup in your hand nervously.

Mandy shook her head violently, her face suddenly pale.

"No. No you can not go back there Y/n. She is dangerous."

"What? No she's not! She's been nothing but nice and has taken care of me."

Your voice shook. Why did Mandy have to be so negative about everything?

"Do you know her?" Mandy asked, her voice sharp and abrasive.

"She's a teacher, she lives alone, she had a daughter once, and she travelled and then settled down here."

Mandy shook her head again.

"No, she's NOT a teacher. She can't be around children after her arrest."

You sat and looked at her, shocked.

"Then, what does she do? She has a big house," you said, looking at the ground. You felt small again. So small.

"THATS WHAT YOURE WORRIED ABOUT? Did you hear me? She was ARRESTED! She didn't travel, she was in jail! Wanna know why, Y/n?"

"No-"

"SHE KIDNAPPED A BABY! She never had a daughter! She tried to steal someone elses! Right from the hospital! Went into the infant ward and plucked a random baby from a cot and tried to stuff it in a bag and run off with it! Thank God that baby was wearing a security bracelet or she could have gotten away."

You slumped in the stool, unable to keep your body up. Your head spun. Nothing felt real all of the sudden and you were sure you were going to be sick. Your eyes stung and you felt your chest tighten. You were drowning.

"No. Th-that can't be true. You're lying."

Mandy huffed and turned back to her laptop, smacking the keys violently. She clicked on something and turned it around to face you, the screen on a newspaper article.

Diane's mugshot was bright and harsh, staring back at you. Right above it was the headline "ATTEMPTED BABY SNATCHING IN ILLINOIS". You opened your mouth, but could find no words. Instead, tears fell from your eyes silently, landing on the table top.

Mandy could see your distress, taking the laptop and shutting it. She reached over and grabbed your hand and caressed it. You looked up at her, your chest aching.

"I'm sorry Y/n, but it's gonna be okay. We're gonna get you some where safe, okay?"

"But-but.. I am safe. That was a long time ago Mandy. She's changed. She's different," she murmured. You couldn't accept what Mandy was saying.

"No honey, she hasn't. I know this about Diane because she was stalking me. She seemed nice at first, but then she started following me. She'd call me nonstop, and just show up to where ever I was. I know she broke into my dorm and stole stuff of mine. Before I started suspecting something was up, I'd go over to her house for dinner and everytime would leave so sick and tired. I could never prove it, but I'm certain she was drugging me."

You shook your head. No, no, absolutely not. You stumbled out of your chair, your head pounding. Your vision was blurring, and you could feel your heart in your throat. You tried to pull in a breath, but your throat was closing. You stumbled and fell, knocking the travel mug onto the floor. You heard it shatter, but it seemed like it happened in another world. Nothing was happening here. It was just you, the cold floor, and your heartbeat.

You heard Mandy say something, running towards the bathroom. You weren't sure what she said, but you didn't try to move. You couldn't move, your body felt like stone. You gasped for air, but you began to choke and gag instead. You were scared, you were sick, and you were heartbroken.

You heard the little ringing of the door's bell in some far off land, and suddenly before your eyes was the face of Diane. You tried to move, and you weren't sure if your goal was to get away from her or to cling to her, but either was pointless. You had no control over your body. You were in one of your panic attacks

All you could do is search her face with your eyes. She looked scared, tears in her eyes, her nose running. She looked to something in the distance, her expression almost scary. She looked back down at you and kissed your forehead before leaning closer to you.

"Hey sweet girl. It's okay. I've got you. I'm gonna take you home now, but I need you to be quiet," she whispered so close to your ear that you could feel her lips on the shell of it.

You wanted to scream, to yell for Mandy, but you could not. You wondered where the coffee shop employees had gone and if they were going to get help. You looked up into Diane's eyes and she smiled desperately at you. Your heart ached. You were so scared, you did not want to go with her. Yet at the same time, part of you did. You could not believe what you had been told.

You felt Diane pick you up from the floor, leaving behind the shattered cup and rushed you out of the door, running down the block with you. She threw you into the passenger seat, not even bothering to buckle you up. You leaned against the door, slumped over. Diane got into the drivers seat and fumbled with the keys, her hand shaking. She turned the ignition and hit the gas, leaving town before Mandy even came back from the bathroom.

Diane drove fast back towards the house, biting her nails. She breathed heavily and you could only watch her out of the corner of your eye. Who was she? Why had she lied? What else was she hiding?

Suddenly Diane took a sharp turn, not breaking, causing you to fly from the seat and fall into her lap. You groaned, feeling a bit more in control. Diane looked down and smiled. Sniffling she petted your hair and shooshed you. You went to push yourself up but Diane pushed you back down. Your breathing became ragged again and you were afraid you were going to hyperventilate.

You felt the car go off road and then go into park. Diane reached behind your seat and got a first aid kit, opening it. Then, you felt a prick in the side of your neck. You gasped and cried out. You scratched at Diane, attempting to push yourself away from her body. Warmth spread through your muscles and your vision seeped away into a black mist, but not before you heard it. Her voice in your ear.

"It's okay baby. I'm here. I'm here now."

#Run to me #run film #sarah paulson #sarah paulson fic #sarah paulson x reader #diane sherman x reader

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oftenderweapons · 4 years

Text

Apple of My Pie — Jin

A Small Town Swoons story

Chapter 1.

Pairing: Kim Seokjin x reader (nicknamed Buttercup)

Wordcount: 3.5k words

Genre: non-idol!AU, Baker/Café owner!Seokjin, University student!reader Flatmates!AU, Friends To Lovers, Fluff, slightest angst.

Rating: suggested 18+ (there are brief apparitions of dirty thoughts, also future episodes will contain NSFW material)

A/N: Hello my sweet poppies! Welcome to the Small Town Swoon Universe! 🥰✨

In this episode: Jin and Buttercup met when she was nothing but a scared, homesick first year student. Four years later, the two share an apartment close to her university and his bakery and café, and are the best of friends, sharing the house, several meals and, most importantly a sacred breakfast ritual. However, as far as sharing goes, Seokjin’s heart has belonged exclusively to Buttercup for four years. Exhausted, Jin finally decides to let go of his unrequited feelings, or at least try.

TRIGGER WARNINGS: Oh, this is chill. Light swearing, heavy infraction of The Silent Roommate Code (aka don’t do the nasty with your bf when your roommate is sleeping in the bed at the other end of the room. Especially if she’s a virgin, first year and very homesick). Also, there is a quick flash image of breast worship, sorry.

Remember to vote for next prompt (check the link in my bio) and in case you need it, here’s my masterlist 💜

In case you need it, here is the music companion

Enjoy! ✨💜

Navi: Chapter 1 — Chapter 2 — Chapter 3 — Chapter 4 — Chapter 5 — Chapter 6 — Chapter 7 (7/7)

It was a slow morning at Jin’s café, only a pair of clients sitting at the small table in the corner, two girls who always met there on Sunday morning, at an illegal hour for the weekend. They had outdoor equipment with them, and probably it was just a stop for a quick breakfast before going skiing or trekking, which was strange considering the disastrous downpour outside, but who was he to debate.

Plus the usual early birds were late too, probably because of the university bonfire the night before.

Seokjin yawned and silently cried over his lost hours of sleep. He was ready to sit down, tip the back of his head against the wall and sleep — actually, rest his eyes —, when the bell at the front door dinged, announcing a new customer.

He inhaled and wore his best smile, standing up. “Good morn— Oh my god, sweetie are you alright?” He asked, seeing a drenched young girl stand at the door.

“I might use a friend.”

That girl was you, running away from your roommate and her boyfriend fucking in your dorm room. Right in the bed beside yours. With you there. And they didn’t even bother keeping quiet.

Seokjin was awestruck. You were soaked like a stray kitten left out in the rain, your hair sticking to your face, your eyes wide and your lip trembling, speaking of several degrees of trauma. “Poor thing.” He murmured, “wait, I should have a blanket back here.”

He dashed for the small cot he had in his office, in the back of the shop, gripping the fleece blanket and bringing it back to the counter, jogging around it and opening the blanket wide as he stared at you. “It’s better if you take off your robe. It’s dripping wet.” He said discreetly.

The girls at the front stared at the scene, a bit worried about you but mostly endeared at the cute barista taking care of you.

“May I use the restroom? The shirt underneath is, uh, thin... Oh, god this is so embarrassing.” You hid your face in your hands.

“Of course,” Jin blushed to his ears, offering you the blanket. “Would you like some coffee? Tea? Cocoa?”

Your lip wobbled, eyes watering and not for the rain. “Cocoa?”

“Yes, sweetie. Go get changed, the restroom is over there.” He pointed at the door.

“Thank you so much.” You said, placing the blanket in front of your chest.

Seokjin rushed behind the counter, grabbing a rag to dry up the wet patches you had left on the floor before someone slipped. Next he got your cocoa ready.

In the quiet morning, through the background music and the gentle chatting of the other two clients, he could hear you using the hand dryer, glad that it was set on hot air so that you could hopefully warm yourself in the process. He even thought of bringing you in the actual bakery, where he had a small traditional stove operated by firewood, other than the big oven working for croissants and banana bread and brownies and pies.

You emerged from the bathroom a little more composed, bundled up in his blanket.

It smelled good. Like raw sugar, butter and apples. A tinge of raisins.

It smelled domestic, like your granny.

You missed your granny.

You missed home.

Your lip wobbled again.

“Come sit”, he said, pointing at a chair in a private corner of the room, somewhere you would be a bit protected from the rest of the shop. It was also conveniently close to the counter, so he could check on you and ask you if you wanted to talk about what had happened. His first thought was that you were a teenage runaway with very bad planning skills, considering that you had run out in your pyjamas and a jacket, your shoes definitely inappropriate for the weather outside, holding only a pair of keys and your wallet in your hands, placing them on the counter once you sat.

“I’m Seokjin.” He said kindly, offering you his hand.

You caught his hand and introduced yourself.

“So, what brings you here with this devil weather so early on a Sunday morning.”

“Running away from my roommate and her boyfriend.” You said, hugging the blanket tighter around you.

“What hap— Nevermind, I think I got it.” Seokjin said, blinking repeatedly. Goodness, people were nasty. “Are you sure you’re okay?” He asked, placing the cup of cocoa in front of you. “Cream? Cocoa powder? Cinnamon? Chocolate sauce? Marshmallows?” He asked.

You teared up. “Marshmallows.”

He poured an abundant amount of them as he pouted, noticing you had become even more upset.

“There you go, Buttercup.” He said, smiling at you so kindly.

“Thank you,” you said, your voice weak and your forehead creased as you desperately tried not to let your tears spill.

“Are you sure you’re okay?” He asked, delicately letting his hands move toward yours, moving slowly to see if you took it away. “May I?” He asked, his fingers hovering over yours.

You nodded. While your left hand held the blanket close to your chest, your right ended pressed between his warm palms, the one on top rubbing your knuckles.

“How old are you?” He asked, worried. He wore a sheepish smile. “I’m sorry, you look very young, I’m just asking to see if I should call your parents or anyone adult.”

“I live at the dorms. I’m in college.” You said, frowning a little.

“As I said, you look very young. And there are some underage students here so...” He explained, his deep, dark eyes breaching through your bad mood.

“I’m a first year. Nineteen.” You said.

“Poor darling, that must be so hard on you.” He said softly, still patting your hand.

You nodded. “I miss my family. My granny.”

“Oh, buttercup.” He cooed.

If you were in a sane state of mind you would have snickered at yourself and at how miserable you looked.

Still, you were grateful for the kind and gentle Seokjin. And how easily he had brought you back home, with the scent of his café, the taste of the cocoa and the specific brand of marshmallow that your grandmother always got for you when you were little.

“It’s a three hour drive. And it’s tough here.” You said, hiding your face as you dried one tear.

“Do you have any friends here?” He asked.

You shook your head. “Not really.”

Seokjin smiled, his eyes becoming even kinder as his cheeks became round and puffy. “From today, I’m your friend.”

Seokjin, you are a strong-willed, honest man. You are a gentleman. You are a good human. He kept repeating in his brain.

You are a polite, friendly, reliable. You are her friend. He repeated as a mantra.

Still, his brain was completely drowned with thoughts of you in the shower.

The two of you had become flatmates in rushed circumstances after you found an apartment ten minutes away from your university, which allowed you to walk there without having to take the bus or end up in the decrepit dorms, sharing a room way too small with someone way too rude or too loud.

Seokjin still didn’t know what had convinced him to share an apartment with you. It was hell. And heaven too, when he didn’t keep reminding himself that you weren’t his girlfriend, that you would never see him like that, and that he shouldn’t be playing house with you.

It was torture and bliss. Bliss on your Sunday mornings, when you could have breakfast together, or random nights when you would have movie marathons together and you would fall asleep against his shoulder, when he would cook for you and you would have dinner together over a glass of wine, laughing and making jokes. The more he spent time with you like that, seeing you drink your morning cup of coffee while still half asleep, on those days when he had someone else doing the morning shift — which was extremely rare — the more he realised you were absolutely perfect for him.

And then torture.

In moments like this, while you were under the shower, when you got out of it and as usual you walked around the house clad in nothing but a towel, absolutely comfortable in your skin, or when you thought he wasn’t home and he could hear your breathy moans and little whimpers, and then again on those two or three nights you had taken somebody home — in those circumstances he felt like he was paying for an ancient crime he didn’t know he had committed.

You had convinced him to move in with you since the apartment — being close to the university — was also incredibly close to his shop, and once he saw your eyes glimmering, your pretty face begging him to accompany you to visit the apartment, he couldn’t really say no.

So, he had said yes.

And once he saw the building, and the warm, domestic ambience, he realised that even if he would never be your lover, the least he could allow himself was to live this small daydream with you.

A week later you and him had signed the papers to rent the place. And everything had escalated from there. You had become the closest of friends, trusting and leaning on each other in every moment, through every difficulty.

However, the more he got to know about the men you dated, the more he realised you would never be attracted to him.

They were all fancy preppy boys who very likely knew the entirety of the Oxford dictionary and could probably recite Shakespeare sonnets impromptu. One of them could easily have been grandson to a duchess or a marquise. And he was pretty sure the first boy you had dated — second year university — had even a trust fund.

It was basically unreal for you to look at him with anything but friendly appreciation.

In an attempt to silence his thoughts, he got out of bed and headed for the kitchen, starting the coffee machine and getting your breakfast ready.

Maybe you would have completely ignored it being January and you would have simply climbed the barstool by the counter wearing your bathrobe, your hair still wet, and the two of you could have had breakfast just like that, without any kind of embarrassment.

As soon as coffee started brewing, your nose appeared from the bathroom door, barely ajar as you slipped out in a soft-looking white t-shirt.

As he threw a glance in your direction he knew immediately that you had very likely stolen the undershirt from his freshly washed laundry.

You slithered out of the bathroom and with stealthy footsteps you occupied your regular spot in the kitchen, watching as he prepared all the necessary material for a respectable breakfast.

“Good morning.” He said as he saw you perched on your favourite seat.

“Morning.” You replied, your feet bare, your toes gripping the small bar connecting the two front legs of the chair. “I thought you were at the café.” You said, pushing your hair away from your face. They weren’t dripping, but they were still a bit damp, especially since you had stopped drying them as soon as the smell of hot coffee reached you in the bathroom.

“Lara is covering the morning shift. I’m doing tea time today. The ladies love me and Lara can’t stand them asking about her boyfriend. I can’t have her kiss and grind on her girlfriend in the middle of my distinguished bakery out of spite.” Jin placed some apple slices on your plate, together with a quite large piece of apple pie.

In a small bowl, he poured some dry fruit before placing it on the table.

“Petty, angsty thing she is.” You said, clicking your tongue. “A true hero.”

He snickered. “Not surprised you’re friends.”

“I am patience made person.” You said, playfully offended.

“Like that one time you smashed a plate on the floor because you had burnt yourself when taking it out of the oven.”

“It was an accident. I dropped it.”

“Like it’s hot.” Seokjin murmured under his breath, lightly swaying his hips as he finished aesthetically placing your food on the plate.

“What?” You asked, comically confused.

“Nothing.” He said, stopping altogether before pouring you some coffee, adding a spray of whipped cream and decorating it with caramel and crushed caramelised almonds.

Jin asked himself how many more times he’d be able to cook you breakfast; how long until he would have to teach someone else, until you would move out with another person and you start your day with crappy industrial food instead of homemade pies and organic apples and his grandmother’s dried hazelnuts and almonds and freshly toasted chestnuts when the season was right.

Whenever he was home, he spoiled you with homemade breakfast. It was the only way he truly allowed himself to show you how desperately in love with you he is. Anytime he cooked, love simply seemed to pour out of his body through the powerful way he kneaded biscuit and pie batter, and the delicate gestures he used to place each part of a dish to form beautiful works of art: crimson red wine risotto on white porcelain plates; juicy cuts of meat, perfectly cooked in that wondrous oven of his, with a deep brown layer on the outside and the most tender dark pink in the middle, laying on the freshest bed of lettuce with a thin dribble of balsamic vinegar and crushed green peppercorn on deep blue rectangular plates.

And every Sunday was sacred. Every Sunday morning he woke up like he had spent all Saturday night courting you and making love to you — minus the obvious relief and satisfaction that come from spending all night on a bed with the person who is your partner and your lover at the same time. Sunday morning was his favourite ritual. Waking you up with the smell of your favourite hot chocolate — the one you seemed to be addicted to, and that he used on you and against you very wisely — and then cake, a different one every week, and again fruit and sometimes, in summer he would go to the closest farm, buy the milk directly from the farmer, a friend of his grandmother, at the crack of dawn on Saturday morning, bring it home, pasteurise it so that it was ready for Sunday morning, when he would use it for the healthiest of smoothies.

He loves you. He has loved you for years. And after two years of living together, losing hope was a possibility.

A possibility a bit too vast at the moment. Actually — hopefully — reality.

Today would be like any other day if it weren’t for one small fact.

Two days ago it had been four years since he first realised he had fallen for you. And two days ago he had decided he would stop chasing after you.

Therefore, he had decided that from then on, he would let go of you, even if that meant losing a part of himself. And today he would actively start walking a new path.

Once the table was ready, he arranged both your and his plate there, without passing you your cup of coffee — as he usually did — and waiting for you to come to the table.

You moved your hair out of the way as you sat down, taking your fork, not even noticing Seokjin’s first sign of petty detachment. You immediately stabbed your fork inside the apple slice and bit into it.

“Do you have lessons today?”

“Romantic Philology in the afternoon.” You replied munching, pushing your hair behind your shoulders, accidentally exposing two wet patches on the front of your t-shirt.

Actually, Seokjin’s t-shirt, but you decided he didn’t need to know that: you had simply forgotten to carry your clothes to the bathroom and once you heard the bustle going on in the kitchen, you managed to find a pair of pyjama pants in the clean laundry, but not a shirt. And you had stolen one of Seokjin’s. Not like it was a big deal.

“Romantic as in love?” He asked.

“No, as in 1830s, German, English and Italian. We’re looking into Byron and Shelley. Sometimes it’s outright boring, but our professor is so hilarious, she sees right through all those pompous arses.” You said, getting started on your masterpiece of a coffee.

“Oh.” Seokjin said. One more point for the preppy kids.

“No, it’s just academic stuff. Nothing that is actually worth something in real life. Some days I just wish I could give up on Goethe and Scott and the Brontes so I could bake cookies without a care in the world.”

And every day he wished he could give you just that. Turn his bakery into your sanctuary, hold you there, half guest, half hostage.

He decided to halt his thoughts there. No more.

“So you have teatime. Do you want me to make dinner tonight?” You asked.

“Actually no.” He said casually.

You stopped munching on your food. “Oh. It’s not Tuesday, though. Are you out with the guys, random meet up? Is Namjoon in town?”

“No.” He glued his eyes to the plate. No, he had not noticed your hardened nipples, a vague halo of dusty pink appearing from underneath the thin, wet white cotton. No. He would not let his mind wander. No, he would smash the thought out of his mind.

Smash you.

No! The thought. His mind. Out.

Like the colour didn’t remind him of fresh raspberry ice cream, like he hadn’t imagined dragging frozen raspberries against your oh-so-responsive buds, only to warm them with his mouth afterwards, pinch the small fruits between his fingers, crush them until tiny droplets of ruby juice landed on your lush breasts, his tongue lashing out to collect the liquid and lave your luscious curves.

But this time the thought did not enter his brain. This time he let it wither and dissolve into fine, sterile dust.

“Are you having dinner with your granny? And you didn’t invite me?” You said, pouting. “Her roast-beef is—” You stopped and swooned. “The definition of perfection.”

“I’m out on a date.” He said briefly and simply.

You frowned and quickly lifted your eyebrows, not letting the confusion show. “You sure you still know how those work?”

“It’s not like I’m celibate.” He said shrugging with his humongous shoulders. Lifting all those sacks of flour… And helping at the farm— You frowned again.

“Cinnamon?” He asked, knowing that the spice sometimes bothered you.

“No, no...”

“Do you need assistance, for your date? You sure you don’t mean the exotic, typically Egyptian fruit?”

“I mean I’m going out with a girl.” Seokjin started growing impatient.

“Who is it?” You asked, out of curiosity. In two years he had never brought a girl home. And in four years you had know each other, you had never seen him with a female friend or an actual girlfriend. You didn’t even know what is his type.

“Her name is Grace. She’s been a regular at the café for a few months now. She asked me out and I thought it would be rude to say no.”

Your interest poked, you placed down your fork. “Did she invite you?” You held your coffee in your hands, trying to keep yourself from gesticulating nervously.

“No. I did.” He said, finishing his pie and starting to eat all the hazelnuts in the small cup.

“I mean. Plenty of girls give you their phone number on a weekly basis. I literally find them everywhere. There’s around thirty on top of the washing machine alone, because I can’t do your laundry and have all those pieces of papers disintegrating and infesting our laundry and the drain. Why didn’t you ignore her like all the rest?” You asked, a bit upset.

“Because she seems a nice person,” who could love me back, which you don’t. He replied, leaving half the motivation silent in his brain.

“Cool.” You said, finishing your coffee before standing up and placing the cup in the sink.

“Cool,” he replied, neutral, watching as you left all the almonds and dried banana slices in the cup, the pie on your plate. “You’re not done with breakfast.”

“I’m late with my homework.” Which you weren’t, but you felt like your breakfast had been poisoned. Maybe that’s why you felt sick in your stomach.

Seokjin pouted and finished his food before placing your leftovers in small boxes. He knew you would come back hungry from uni and finish the food you had abandoned.

He didn’t read too much into your reaction. He was done trying to understand you.

Today he was finally done being stuck at a crossroad, and although your path in the woods felt and looked lovely and smelled even better, he opted for the safe, trodden and charted way that led out of the woods, into the uneventfulness of the ordinary.

———————————————————

Navi: Chapter 1 — Chapter 2 — Chapter 3 — Chapter 4 — Chapter 5 — Chapter 6 — Chapter 7 (7/7)

#bangtanhq #thetruthuntoldnet #kim seokjin x reader #Jin x reader #kim seokjin #Baker!seokjin #bts fanfiction #bts blog #seokjin fluff #Seokjin flatmates!Au #Seokjin friends to lovers

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bbrandy2002 · 5 years

Text

Birth of an Heir... Part Two

Book: The Royal Heir

Liam x Riley

Part one

A/N: Turned into another total fluff piece, I'm becoming a softy 😕. I had one of my kids choose the baby's name of the ones given in the comments and @kingliam2019 was the winner.

Warning: Drake Walker has a potty mouth and will use profanity in the chapter, but, he's still lovable marshmallow all the same.

These characters belong to Pixelberry.

Birth of an Heir Part 2

Riley stands at the window of her third floor hospital room, observant of the overflowing press crowded below. It had been several hours since the birth of her son and well past midnight. Exhaustion had reached it's peak, yet, adrenaline and a newborn would keep her awake. Another news crew ascends into the mass influx of cameras, sateliites and curious, Cordonian bystanders. "Do they ever sleep?", she asks with astonishment, however, her question is met with silence. She turns, and instantly her heart melts from a simple, yet, poignant sight.

"You are hopeless, you know that right?", she says with a gentle smile.

Liam is sitting in a rocking chair, holding his first born child, whom seems to fit perfectly, like a small package in his arms. His thumb grazing the soft cheeks and his eyes focused, contently, on what he considers the second best thing to enter his life. With heavy eyes, he searches and memorizes every single line and detail, and takes to heart, every sound and breath this tiny creature makes. He is afraid to fall asleep, he cannot fathom for one moment the seperation that comes with it. This baby already has his father under a spell that is similar to the way his mother does.

Riley walks closer to him, still slowed by soreness and tenderness with each step. She shakes her head and playfully rolls her eyes, "Liam, did you hear what I said?"

He looks up to her, "I'm sorry dear, did you need something?".

She bites her bottom lip before lightly rubbing her hand up and down his arm, "I said you're hopeless....you know I have to hold him more than just when feeding him, right?".

Liam nods his head and speaks with a teasing whisper, "We'll see about that", as he gives her a quick wink.

Riley sighs, "yep.....hopeless", then leans in to a soft kiss.

She removes her robe, kicks off her slippers, then lowers herself steadily onto her hospital bed. After pulling the sheet over her tired and worn body, out of habit, she rolls to her left side. Riley wants to drift off, but, the events of the day are flourishing in her thoughts. Suddenly, like a vision of sorts, one that is always present in her mind, a familiar story begins to play......

A flash of bright blue that meets her soft brown. An instant spark, a physical attraction, a moment when two lost souls find one another. A beginning.

"Sorry I'm late. Thank you for your patience, Miss...?"

The first time the heat of his lips devoured hers after an impromtu private tour of the Statue of Liberty.

"You're full of surprises aren't you?"

"I try"

"I'm glad to have met you Riley, I'll never forget this night".

A chance of a lifetime, when fate opened a door and offered her an opportunity.

"You want to sponsor me? Why?".

"I'm not doing it just for you, I saw how Liam looked at you last night. I've never seen him so happy...".

The first time he swept her off her feet, literally....the moment her heart felt more than just an attraction, but, the start of a love affair. A waltz to remember.

"I dont....I'm not sure I know the steps."

"Just follow my lead".

Like something out of a fairy tale, darkness surrounded by small, white lights and the shimmer of the moon. Closed off from the rest of the world, it would become a place where memories are made.

"We made it....the center of the maze..and I do believe I won".

"How do you figure that?".

"Easy, I'm on top".

"Hard to argue with that".

"Then don't"

"Something about you...it just feels...right to be around you".

"Liam, I feel the same way".

An exhilerating jump, involving total trust in one another.

"It's said that if two lovers climb to the top of the waterfall and jump in together, they'll get the blessings of the water".

A warm, spring day, filled with pink blossoms and the aroma of freshly baked pies. When she became a queen for the first time and the apple of his eye.

"Meeting you in New York was one of the best things that ever happened to me. You changed my life".

A country soiree, a swing decorated with flowers. He was giddy, playful and childlike.

"...I want you to know, I can't wait for the coronation".

"I can't wait either".

The simple, yet, perfect candlelit dinner, when he opened up to her and shared his biggest dream.

"Maybe this is sentimental, but, its been on my mind lately.....What I really want is to have a family...and not the type most kings have, but, one where we're close and listen to each other....".

The night the stars glistened in the sky above her, three words that rained down and changed everything. A merging of pleasure and senses like never before. The most beautiful , endearing intertwining of bodies and demonstration of devotion ever.

"I've never said this before, but, its more true than my own name. Which is all to say...Riley...I love you".

"Liam, I've been madly in love with you since the day we met...".

"I love you and I want you".

"Then take me, Liam".

And after a period of turmoil, secrecy , and undeserved scrutiny, when she realized air doesn't exist without Liam....she doesn't exist without Liam....they came full circle.

"Riley...all that I want in this life is to dedicate my life to being the best man and king that I can be...for you...Lady Riley Brooks, queen of my heart, I have yearned to say these four words for a very, very long time...Will you marry me?".

"Yes...a thousand times...yes".

When happily ever after truly began....

"I promise to love you, honor you and cherish you. To support you, fight for you and inspire you....as you have always done for me....No matter what the future holds for us, I know that together is where we're meant to be".

A warmth overcomes her body and the corner of her lips curl. She opens her eyes again and peaks once more at the two most sacred things in her life. One is her soulmate, her king, her love; the other is their finest masterpiece.

A faint cry echo's through the calmness of the darkened room. Liam gently lifts the baby to his chest, its head nuzzled securely into his neck. He places one hand on its bottom and the other lightly presses soft chubby cheeks closer to him. He leans forward, planting a lingering kiss through a blue cap, covering fine, peach fuzz hairs. Liam senses her stare and glances up. A faint smile emerges, that not a human in Cordonia could wipe away. With a low, raspy voice, he utters, "I love you".

Those words always send a chill down her spine, however, the emotion and pride in his voice this time, causes her to shed a tear, "I love you, too".

Her body gives way to sleep, assured that her two boys will keep one another company until she wakes.

************

Eyes slowly flutter open and Riley recognizes the presence of early dawn's light. She lifts her stiffened neck to see Liam laying in a small bed near the window; when you're the King, the hospital ensures you continue to sleep like royalty. Next to his bed, he had slid the baby's cot next to him.

She rubbed her eyes, trying to fully awaken. After getting out of her bed, she slips her shoes back on and treks to her infants side. Liam actually fell asleep with his hand in the cot, so Riley picks it up and moves it, placing it on his chest, -Hopeless.

She then lifts her wiggling bundle up and carries him back to her bed.

Lowering her gown down, the baby latches on after a few tries, suckling with tiny fingers gripped to her breast.

Riley watches in awe and amazement, as the most precious bond in the world takes shape. His little eyes twinkle and he is full of energy. She chuckles at his arm movements and how he seems to be devouring his breakfast.

She smiles at him, then tilts her head; she couldn't tell last night, but, relishes in the fact that he definately looks every bit like her husband.

There is a tap at the door and Riley gives permission for them to enter. The first thing she sees is a cascade of balloons and a large stuffed teddy bear, followed by the always gleeful, Maxwell. Trailing behind him, a seemingly recovered Hana, and weary eyed, Drake.

Once Hana slaps a few balloons from her face, she makes her way over to Riley's bedside. "Oh my goodness...Riles, he's so small and handsome".

After Drake gets caught up in the balloons as well and Maxwell runs into him a few times with the stuffed bear, he gets another glance at his namesake. Realizing she is breastfeeding, his face begins to blush, he stammers and fidgets. "Um...um...yeah...I'm just going to...um...look at this wall...that's some fine craftsmanship...this wall". He starts knocking on the wall as if he is checking the quality of its structure.

Maxwell laughs, "Augh, come on Drake, its just boobies....besides, the kids head is blocking the good stuff."

Hana and Riley look at each other, both chuckling at the two men.

Riley notices that Bertrand was not with them and asks Maxwell if he plans to visit.

Maxwell and Hana look at each other and snorted out in laughter.

Confused, "What's so funny guys...what am I missing?", she asks.

Drake still facing the wall, himself amused by the answer to her question, can't help but chime in, "Let just say, ol' Bertrand....", he starts laughing again, "Bertrand....". Drake can"t get the words out before he starts snickering even harder.

Annoyed by all the laughing, Riley threatens to kick all of them out unless they tell her what is so funny.

All three make attempts, however, are still struggling; they have tears running down their cheeks and faces that are bright red.

"That's it...out", Riley finally tells them.

Drake is able to compose himself, taking in deep breaths, "Okay, so...last night at the shower...everyone got...well..got sick from Maxwell's fucking punch as you know", he continues with some difficulty in spitting the words out.

Maxwell wipes a tear from his eye, "Let's just say, Bertrand had to call someone about the mess Madeleine made off the balcony last night".

Riley's grimaces, "Okay. I don't even want to know".

Hana asks if the baby has a name yet.

She looks down at her son, who has finished his feeding and covers herself back up. Riley adjusts the baby into a more comfortable position and looks up with a smile. "I would like to introduce you to the new Crown Prince of Cordonia...Nikolas Drake Beaumont Lee".

All three are touched as she explains how important each one of them are to her and Liam, and hopes the baby carries a little pieces of his namessake. After a short visit, her friends leave, Maxwell promising to return later with junk food and booze.

Shortly after, Liam stirs, surprised she didn't wake him up.

He shuffles to her bedside, kissing her tenderly. Liam searches her eyes, cupping her cheek, "I just want you to know, you are the most incredible woman I have ever met. You have always amazed me, but, I will never be able to thank you enough for making all of my dreams come true".

He has a way with words, always has, she thinks to herself. She gets a mischievous smile, "I know a way that you can start thanking me".

He raises an eyebrow, intriqued, "Anything for my queen".

She lifts the baby up and gently places him in Liam's loving arms, "He needs a diaper change".

@ao719 @carabeth @emceesynonymroll @romanticatheart-posts @kimmiedoo5 @ladyangel70 @polishchoicesfan @zorjasblog @burnsoslow

#choices trr #the royal heir #trh #choices trh #liam x mc #liam x riley #maxwell beaumont #trr drake #trr fanfic #trr liam #trr #the royal romance #drake walker

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worldpedias · 3 years

Text

How To Integrate Sin^2 <3

Let’s start off with an integral that we should already be able to do. X function with respect to x is equal to sum of the negative cos.

Integral of sin^2(nx)/sin^2(x) from 0 to pi YouTube from www.youtube.com

How to integrate sin^2. Integrate 1/(cos(x)+2) from 0 to 2pi; For each of these, we simply use the fundamental of calculus, because we know their corresponding derivatives. The integration of sine inverse is of the form.

Here are some examples illustrating how to ask for an integral. , csc cot, sec tan, csc. ∫ cosxsin5xdx = ∫ u5du using the substitution u =sinx = 1 6 sin6x+c ∫ cos.

Z = 2 π x, then. These is special integral fresnel integral. We have to integrate of sin 2 x.

We intend to travel a simple path from 0 to x, but we end up with a smaller percentage instead. Our calculator allows you to check your solutions to calculus exercises. = s(u) plug in solved integrals:

D x = π 2 d z, and. S (z) = ∫ sin ⁡ (π z 2 2) d z (1) let. The integral of sin(x) multiplies our intended path length (from 0 to x) by a percentage.

We recall the pythagorean identity and rearrange it for cos 2 x. X function with respect to x is written in the following mathematical form in calculus. All common integration techniques and even special functions are supported.

#wallpaper #integrate

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mathematicianadda · 4 years

Text

Advanced Integral: $\int_0^1\frac{\text{Li}_2(x^2)\arcsin^2(x)}{x}dx$ https://ift.tt/eA8V8J

How to tackle

$$I=\int_0^1\frac{\text{Li}_2(x^2)\arcsin^2(x)}{x}dx\ ?$$

This integral came up while I was working on some harmonic series.

First attempt: By writing $\text{Li}_2(x^2)=-\int_0^1\frac{x^2\ln(y)}{1-x^2y}dy$ we have

$$I=-\int_0^1\ln(y)\left(\int_0^1\frac{x\arcsin^2(x)}{1-x^2y}dx\right)dy$$

and Mathematica gave a complicated expression for the inner integral and that made me stop.

Second attempt: $x=\sin\theta$

$$I=\int_0^{\pi/2}\theta^2\cot\theta\ \text{Li}_2(\sin^2\theta)d\theta$$

$$=\sum_{n=1}^\infty\frac{1}{n^2}\int_0^{\pi/2}\theta^2\cot\theta \sin^{2n}(\theta) d\theta$$

and I have no idea how to continue. Any suggestion?

Thanks

from Hot Weekly Questions - Mathematics Stack Exchange Ali Shather from Blogger https://ift.tt/327xlz0

#Blogger

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daotaolienthong · 5 years

Text

Cách học Công thức lượng giác chuẩn & đơn giản bằng “Thơ”

Công thức lượng giác là một trong những kiến thức không quá khó nhưng cũng không dễ chút nào. Công thức lượng giác cơ bản gồm nhiều phương trình thể hiện mối liên hệ với nhau giữa các hàm sin, cos… là công cụ để bạn giải quyết các bài toán liên quan đến lượng giác. Ngoài ra, để vận dụng được các công thức lượng giác bạn phải hiểu được bản chất vấn đề và nắm rõ các kỹ năng cần thiết. Do đó việc đầu tiên các bạn cần phải làm đó là học những kiến thức cơ bản để nắm chắc vấn đề.

[caption id="attachment_5183" align="alignnone" width="1087"] Công thức lượng giác[/caption]

Lịch sử Lượng giác

Lượng giác (tiếng Anh: Trigonometry) là một nhánh toán học dùng để tìm hiểu về hình tam giác và sự liên hệ giữa cạnh của hình tam giác và góc độ của nó.

Nguồn gốc của lượng giác được tìm thấy trong các nền văn minh của người Ai Cập, Babylon và nền văn minh lưu vực sông Ấn cổ đại từ trên 3000 năm trước. Các nhà toán học Ấn Độ cổ đại là những người tiên phong trong việc sử dụng tính toán các ẩn số đại số để sử dụng trong các tính toán thiên văn bằng lượng giác. Lagadha là nhà toán học duy nhất mà ngày nay người ta biết đã sử dụng hình học và lượng giác trong tính toán thiên văn học trong cuốn sách của ông Vedanga Jyotisha, phần lớn các công trình của ông đã bị tiêu hủy khi Ấn Độ bị người nước ngoài xâm lược.

Nhà toán học Hy Lạp Hipparchus vào khoảng năm 150 TCN đã biên soạn bảng lượng giác để giải các tam giác.

Một nhà toán học Hy Lạp khác, Ptolemy vào khoảng năm 100 đã phát triển các tính toán lượng giác xa hơn nữa.

Nhà toán học người Silesia là Bartholemaeus Pitiscus đã xuất bản công trình có ảnh hưởng tới lượng giác năm 1595 cũng như giới thiệu thuật ngữ này sang tiếng Anh và tiếng Pháp.

Bảng công thức lượng giác đầy đủ

Giá trị lượng giác của một góc

[caption id="attachment_5174" align="aligncenter" width="462"] Giá trị lượng giác của một góc[/caption]

Cung liên kết (Cos đối, Sin bù, phụ chéo)

[caption id="attachment_5175" align="aligncenter" width="573"] Cung liên kết (Cos đối, Sin bù, phụ chéo)[/caption]

Công thức lượng giác cơ b��n

[caption id="attachment_5176" align="aligncenter" width="322"] Công thức lượng giác cơ bản[/caption]

Công thức cộng

[caption id="attachment_5177" align="aligncenter" width="490"] Công thức cộng[/caption]

Công thức nhân đôi, nhân ba

[caption id="attachment_5178" align="aligncenter" width="632"] Công thức nhân đôi, nhân ba[/caption]

Công thức hạ bậc

[caption id="attachment_5179" align="aligncenter" width="488"] Công thức hạ bậc[/caption]

Công thức biến đổi tổng thành tích

[caption id="attachment_5180" align="aligncenter" width="554"] Công thức biến đổi tổng thành tích[/caption]

Công thức biến đổi tích thành tổng

Cách học công thức lượng giác bằng thơ

Công thức CỘNG trong lượng giác

Cos + cos = 2 cos cos

cos trừ cos = trừ 2 sin sin

Sin + sin = 2 sin cos

sin trừ sin = 2 cos sin.

Sin thì sin cos cos sin

Cos thì cos cos sin sin “coi chừng” (dấu trừ).

Tang tổng thì lấy tổng tang

Chia một trừ với tích tang, dễ òm.

HÀM SỐ LƯỢNG GIÁC

Bắt được quả tang

Sin nằm trên cos (tan@ = sin@:cos@)

Cotang dại dột

Bị cos đè cho. (cot@ = cos@:sin@)

Bắt được quả tang

Sin nằm trên cos

Côtang cãi lại

Cos nằm trên sin!

GIÁ TRỊ LƯỢNG GIÁC CỦA CÁC CUNG ĐẶC BIỆT

Cos đối, sin bù, phụ chéo, khác pi tan

Cosin của 2 góc đối bằng nhau; sin của 2 góc bù nhau thì bằng nhau; phụ chéo là 2 góc phụ nhau thì sin góc này = cos góc kia, tan góc này = cot góc kia; tan của 2 góc hơn kém pi thì bằng nhau.

CÔNG THỨC LƯỢNG GIÁC NHÂN BA

Nhân ba một góc bất kỳ, sin thì ba bốn, cos thì bốn ba, dấu trừ đặt giữa 2 ta, lập phương chỗ bốn, … thế là ok.

Công thức gấp đôi:

+Sin gấp đôi = 2 sin cos +Cos gấp đôi = bình cos trừ bình sin = trừ 1 + 2 lần bình cos = + 1 trừ 2 lần bình sin +Tang gấp đôi Tang đôi ta lấy đôi tang (2 tang) Chia 1 trừ lại bình tang, ra liền.

Cách nhớ công thức: tan(a+b)=(tan+tanb)/1-tana.tanb

tan một tổng 2 tầng cao rộng trên thượng tầng tan + tan tan dưới hạ tầng số 1 ngang tàng dám trừ một tích tan tan oai hùng

CÔNG THỨC LƯỢNG GIÁC BIẾN ĐỔI TÍCH THÀNH TỔNG

Cos cos nửa cos-+, + cos-trừ Sin sin nửa cos-trừ trừ cos-+ Sin cos nửa sin-+ + sin-trừ

CÔNG THỨC LƯỢNG GIÁC BIẾN ĐỔI TỔNG THÀNH TÍCH

sin tổng lập tổng sin cô cô tổng lập hiệu đôi cô đôi chàng còn tan tử + đôi tan (hoặc là: tan tổng lập tổng 2 tan) một trừ tan tích mẫu mang thương sầu gặp hiệu ta chớ lo âu, đổi trừ thành + ghi sâu vào lòng

Một phiên bản khác của câu Tan mình + với tan ta, bằng sin 2 đứa trên cos ta cos mình… là

tanx + tany: tình mình + lại tình ta, sinh ra 2 đứa con mình con ta

tanx – tan y: tình mình hiệu với tình ta sinh ra hiệu chúng, con ta con mình

CÔNG THỨC CHIA ĐÔI (tính theo t=tg(a/2))

Sin, cos mẫu giống nhau chả khác Ai cũng là một + bình tê (1+t^2) Sin thì tử có 2 tê (2t), cos thì tử có 1 trừ bình tê (1-t^2).

HỆ THỨC LƯỢNG TRONG TAM GIÁC VUÔNG

Sao Đi Học (Sin = Đối / Huyền) Cứ Khóc Hoài ( Cos = Kề / Huyền) Thôi Đừng Khóc ( Tan = Đối / Kề) Có Kẹo Đây ( Cotan = Kề/ Đối)

Sin : đi học (cạnh đối – cạnh huyền) Cos: không hư (cạnh đối – cạnh huyền) Tang: đoàn kết (cạnh đối – cạnh kề) Cotang: kết đoàn (cạnh kề – cạnh đối)

Tìm sin lấy đối chia huyền Cosin lấy cạnh kề, huyền chia nhau Còn tang ta hãy tính sau Đối trên, kề dưới chia nhau ra liền Cotang cũng dễ ăn tiền Kề trên, đối dưới chia liền là ra

Sin bù, cos đối, hơn kém pi tang, phụ chéo. +Sin bù :Sin(180-a)=sina +Cos đối :Cos(-a)=cosa +Hơn kém pi tang : Tg(a+180)=tga Cotg(a+180)=cotga +Phụ chéo là 2 góc phụ nhau thì sin góc này = cos góc kia, tg góc này = cotg góc kia.

Công thức tổng quát hơn về việc hơn kém pi như sau: Hơn kém bội 2 pi sin, cos Tang, cotang hơn kém bội pi. Sin(a+k.2.180)=sina ; Cos(a+k.2.180)=cosa Tg(a+k180)=tga ; Cotg(a+k180)=cotga *sin bình + cos bình = 1 *Sin bình = tg bình trên tg bình + 1. *cos bình = 1 trên 1 + tg bình. *Một trên cos bình = 1 + tg bình. *Một trên sin bình = 1 + cotg bình. (Chú ý sin *; cos @ ; tg @ ;cotg * với các dấu * và @ là chúng có liên quan nhau trong CT trên)

Học công thức lượng giác “thần chú”

• Sin= đối/ huyền

Cos= kề/ huyền

Tan= đối/ kề

Cot= kề/ huyền

* Thần chú: Sin đi học, Cos không hư, tan đoàn kết, cotan kết đoàn

Hoặc: Sao đi học, cứ khóc hoài, thôi đừng khóc, có kẹo đây!

• Công thức cộng:

Cos(x y)= cosxcosy sinxsiny

Sin(x y)= sinxcosy cosxsiny

* Thần chú: Cos thì cos cos sin sin

Sin thì sin cos cos sin rõ ràng

Cos thì đổi dấu hỡi nàng

Sin thì giữ dấu xin chàng nhớ cho!

Tan(x+y)=

* Thần chú: Tan một tổng hai tầng cao rộng

Trên thượng tầng tan cộng cùng tan

Hạ tầng số 1 ngang tàng

Dám trừ đi cả tan tan oai hùng

Hoặc: Tang tổng thì lấy tổng tang

Chia một trừ với tích tang, dễ òm.

• Công thức biến đổi tổng thành tích:

Ví dụ: cosx+cosy= 2cos cos

(Tương tự những công thức như vậy)

* Thần chú: cos cộng cos bằng 2 cos cos

Cos trừ cos bằng – 2 sin sin

Sin cộng sin bằng 2 sin sin

Sin trừ sin bằng 2 cos sin.

* Tan ta cộng với tan mình bằng sin hai đứa trên cos mình cos ta.

Công thức biến đổi tích thành tổng:

Ví dụ: cosxcosy=1/2[cos(x+y)+cos(x-y)] (Tương tự những công thức như vậy)

* Thần chú: Cos cos nửa cos-cộng, cộng cos-trừ

Sin sin nửa cos-trừ trừ cos-cộng

Sin cos nửa sin-cộng cộng sin-trừ.

• Công thức nhân đôi:

Ví dụ: sin2x= 2sinxcosx (Tương tự những công thức như vậy)

Thần chú: Sin gấp đôi = 2 sin cos

Cos gấp đôi = bình cos trừ bình sin

= trừ 1 cộng hai bình cos

= cộng 1 trừ hai bình sin

Chỉ việc nhớ công thức nhân đôi của cos bằng thần chú trên rồi từ đó có thể suy ra công thức hạ bậc.

Tang gấp đôi=Tang đôi ta lấy đôi tang (2 tang)

Chia 1 trừ lại bình tang, ra liền.

• Hàm số lượng giác và các cung có liên quan đặc biệt:

Ví dụ: Cos(-x)= cosx

Tan( + x)= tan x

* Thần chú: Sin bù, Cos đối,Tang Pi,

Phụ nhau Sin Cos, ắt thì phân chia

Hoặc: Cos đối, sin bù, phụ chéo, hơn kém pi tang .

Kết

Trên đây là tất cả những công thức lượng giác và cách ghi nhớ chúng mà dean2020.edu.vn muốn chia sẻ với các bạn. Công thức nào cũng cần thiết và phương pháp ghi nhớ nào cũng hữu ích nhưng yếu tố cốt lõi vẫn là ở chính bản thân bạn. Sẽ không thể nhìn hay nghe một lần đã đã có thể ghi nhớ và vận dụng, vậy nên hãy siêng năng chăm chỉ luyện tập nhé. Chúc các bạn học tập tốt.

Coi bài nguyên văn tại : Cách học Công thức lượng giác chuẩn & đơn giản bằng “Thơ”

#dean2020

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universallyladybear · 5 years

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De chasse dans le morvan secteur glux en glenne à cheval sur deux départements 71-58 nous chassons le sanglier le chevreuil plus…

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Dans le 06 ou le 04 juin 2019 nous contacter pour plus d’informations tel au 06 75 75 36 62 publiée le 3 juin 2019 actions de chasse disponibles.

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Gros gibiers 3 juin un groupe sur une propriété en sologne dans le cher de 50ha pouvant accueillir 2 fusils à chaque soirée prix 60€ le poste pour un chasseur.

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La recherche d’actions de chasse à la journée | alex0174 | 20 mai 2019 je recherche pour la saison 2019 2020 postés actionnaires ou demi actionnaires traqueurs petits ou grands chiens prix.

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De couleur bleue lof nés le 29 avril 2019 société de 140 ha dans le sud-ouest de la france et pour le plaisir de la journée vers.

Grands gibiers étudie toute proposition raisonnable publiée le 20 juin 2019 nous recherchons 4 actionnaires sérieux respectueux du gibier et des règles de sécurité publiée le. Chien de pied mène et rapproche tous deux il sera pucé vacciné vermifugée et primo vaccinée cherche sa famille dans le var.nous. € publiée le 5 juin 2019 pour la le plaisir de nombreux chasseurs après avoir commandé vous avez 14. 2 fusils maxi étudie toutes propositions publiée le 17 juin 2019 bonjour nous sommes une petite équipe de chasseurs à l’arc je chasse seul sans accompagnateur ou éventuellement l’approche.

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Le Bon Coin Chasse De chasse dans le morvan secteur glux en glenne à cheval sur deux départements 71-58 nous chassons le sanglier le chevreuil plus...

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devphilamaths · 3 years

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Solve sin(pi/2+theta) | sin(pi/2+ x) | sin pi/2 + x formula, Find value sin pi by 2 + x

#Solve sin(pi/2+theta)#sin(pi/2+ x)#sin pi/2 + x formula #Find value sin pi by 2 + x #sin #cos #tenerife #co #cot #sec #cosec #math #12th class

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vieclam123 · 5 years

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Đạo hàm là gì? Công thức tính đạo hàm cần biết

Tất cả những thông tin về đạo hàm sẽ giúp bạn hiểu và sử dụng tốt hơn trong khi học hay trong thực tiễn. Hãy cùng tham khảo nhé.

1. Thế nào là đạo hàm?

Đạo hàm là tỉ số giữa số gia của hàm số với số gia của đối số tại điểm $x_0$, khi số gia đối số tiến sát đến 0 chính là đạo hàm của hàm $y=f(x)$ tại $x_0$.

Đạo hàm của hàm số $y=f(x)$ ký hiệu là $y′(x_0)$ hoặc $f′(x_0)$.

$f'(x_0)=\lim\limits_{ \Delta x \to 0} \frac{ \Delta y}{ \Delta x}$ hoặc $f'(x_0)=\lim\limits_{\Delta x \to 0} \frac{f(x)-f(x_0)}{x-x_0}$

Lưu ý:

• Số gia của hàm số: $\Delta y=y-y_0$

• Số gia của đối số: $\Delta x=x-x_0$

Có nghĩa là: Đạo hàm bằng $\Delta y$ chia $\Delta x$. Trong đó, $\Delta x$ có giá trị vô cùng nhỏ.

Giá trị của đạo hàm tại 1 điểm $x_0$ sẽ có ý nghĩa:

• Thể hiện chiều biến thiên của hàm số đang tăng hay giảm, xem đạo hàm tại đó dương + hay âm –

• Thể hiện độ lớn của biến thiên của hàm số. Nếu đạo hàm = 1 suy ra $\Delta y$ tăng bằng $\Delta x$.

Đạo hàm một bên

Đạo hàm có đạo hàm một bên là bên trái hoặc bên phải. Cụ thể:

• Đạo hàm bên trái của hàm số khi Δx tiến dần đến $0^-$ (nghĩa là $x$→$x_0$ và nhỏ hơn $x_0$): y = f(x) tại $x_0$ được ký hiệu là $f'(x_0^-)$

• Đạo hàm bên phải của hàm số khi Δx tiến dần đến $0^+$ (nghĩa là $x$→$x_0$ và lớn hơn $x_0$): y = f(x) tại $x_0$ ký hiệu là $f'(x_0^+)$

• y = f(x) có đạo hàm tại điểm $x_0$ $\Leftrightarrow f'(x_0)=f'(x_0^-)=f'(x_0^+)$

2. Ý nghĩa và ứng dụng đạo hàm

Đạo hàm cho thấy tốc độ thay đổi của đại lượng đó khi có sự thay đổi và tốc độ thay đổi nhanh hay chậm. Do đó, đạo hàm có thể dùng như một công cụ quan trọng về sự thay đổi diễn ra như thế nào mọi lúc mọi nơi.

Đạo hàm dương khi hàm số đang tăng, tốc độ tăng càng nhanh, đạo hàm càng lớn. Đạo hàm âm khi hàm số đang giảm, theo đó hàm số giảm càng nhanh thì âm càng nhiều.

Ứng dụng vào thực tiễn, đạo hàm có thể cho bạn biết tốc độ tăng trưởng kinh tế để ứng dụng đầu tư vào chứng khoán tốt nhất hay biết về tốc độ gia tăng dân số cho từng vùng cụ thể. Xác định tốc độ phản ứng hóa học, gia tốc của chuyển động, tính toán tốc độ. Để có kết quả, bạn cần có hàm số mô tả đại lượng để tìm đạo hàm của điều mình quan tâm.

Xem hàm số có giá trị lớn nhất và nhỏ nhất ở đâu để tối ưu các hoạt động trong cuộc sống. Khi hàm số đạt giá trị cực đại khi đó đạo hàm bằng 0, lưu ý có ngoại lệ. Từ đó, có thể biết đại lượng có giá trị lớn nhất và nhỏ nhất ở đâu để tối ưu hóa theo mong muốn đề ra. Ví dụ một công ty tính số sản phẩm nên sản xuất để đạt lợi nhuận cao nhất từ đây. Hay tính sao cho hộp sữa có nhiều sữa nhất bằng cách này với nguyên liệu có sẵn… áp dụng thiết kế một lon nước ngọt cũng tương tự.

3. Quy tắc của đạo hàm

Quy tắc cơ bản của tính đạo hàm

Quy tắc đạo hàm của hàm số hợp

Nếu y = y(u(x)) thì y'(x) = y'(u) * u'(x)

4. Đạo hàm gồm những công thức cơ bản gì?

5. Chi tiết công thức của đạo hàm lượng giác

$(\sin (x))'=\cos (x)$

$(\cos (x))'=-\sin (x)$

$(\tan (x))'=(\frac{\sin (x)}{\cos (x)})'=\frac{\cos ^2(x)+\sin^2(x)}{\cos^2(x)}=\frac{1}{\cos^2(x)}=sec^2(x)$

$(\cot(x))'=(\frac{\cos (x)}{\sin (x)})'=\frac{-\sin^2(x)-\cos^2(x)}{\sin^2(x)}=-(1+\cot^2(x))=-csc^2(x)$

$(sec(x))'=(\frac{1}{\cos (x)})'=\frac{\sin (x)}{\cos^2(x)}=\frac{1}{\cos (x)}.\frac{\sin (x)}{\cos (x)}=sec(x) \tan (x)$

$(csc(x))'=(\frac{1}{\sin (x)})'=-\frac{\cos (x)}{\sin^2(x)}=-\frac{1}{\sin (x)}.\frac{\cos (x)}{\sin (x)}=-csc(x)\cot(x)$

$(arcsin(x))'=\frac{1}{\sqrt{1-x^2}}$

$(arccos(x))'=\frac{-1}{\sqrt{1-x^2}}$

$(arctan(x))'=\frac{1}{x^2+1}$

6. Bảng đạo hàm là gì?

7. Bài tập tính đạo hàm

Sau đây là cách làm bài tập tính đạo hàm để bạn tham khảo kỹ năng cho bản thân nhé.

7.1. Đơn giản hóa chức năng của hàm

Để đơn giản hóa chức năng của đạo hàm sao cho vẫn mang lại cùng một đạo hàm nhưng thay vì khó tính toán, bạn sẽ thực hiện tính toán đơn giản hơn nhiều. Ví dụ có phương trình (6x + 8x) / 2 + 17x +4, bạn thực hiện đơn giản hóa theo các đơn giản hóa như sau:

• (14x) / 2 + 17x + 4

7x + 17x + 4

=> 24x + 4

7.2. Xác định dạng của hàm

Tìm hiểu các hình thức khác nhau như:

• Là một số cụ thể như số 4

• Gồm 1 số nhân với một biến không có số mũ như 4x

• Gồm 1 số nhân với một biến có số mũ (ví dụ 4x ^ 2)

• Hay hình thức 4x + 4

• Nhân các biến dạng x * x

• Hình thức phân chia biến dạng x / x

7.3. Một số

Đạo hàm của một hàm ở dạng này luôn có giá trị bằng 0. Ví dụ:

• (4) '= 0

• (-234059) '= 0

• (pi) '= 0

Lưu ý: Kết quả này xảy ra là do không có sự thay đổi trong hàm. Theo đó, giá trị của hàm sẽ luôn là số mà đề bài cung cấp trước.

7.4. Một số nhân với một biến không có số mũ

Đạo hàm của một hàm ở dạng này luôn là số. Ví dụ:

• (4x) '= 4

(x) '= 1

(-23x) '= -23

Lưu ý: Hàm sẽ tăng với tốc độ ổn định, không đổi, không thay đổi nếu x không có số mũ. Từ phương trình tuyến tính y = mx + b sẽ giúp bạn nhận ra thủ thuật này.

7.5. Một số nhân với một biến có số mũ

Thực hiện công thức tính đạo hàm này, ta có:

• Nhân số với giá trị của số mũ

• Trừ một từ số mũ

Ví dụ:

• (4x ^ 3) '= (4 * 3) (x ^ (3-1)) = 12x ^ 2

(2x ^ 7) '= 14x ^ 6

(3x ^ (- 1)) '= -3x ^ (- 2)

Ghi chú: Các ví dụ có các ghi chú: biểu tượng cho một đạo hàm là biểu tượng ', dấu * để nhân, dấu ^ là số mũ.

Hy vọng những thông tin về đạo hàm và công thức tính đạo hàm ở trên đã mang tới cho bạn những thông tin bổ ích cho việc học hay áp dụng vào cuộc sống.

Tham khảo bài gốc ở: Đạo hàm là gì? Công thức tính đạo hàm cần biết

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devphilamaths · 3 years

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Solve sec(pi/2+theta) | sec(pi/2 +x) | sec pi/2 + x formula, Find Exact value sec pi by 2 + x

#Solve sec(pi/2+theta)#sec(pi/2 +x)#sec pi/2 + x formula #Find Exact value sec pi by 2 + x #sec(pi/2+theta)#se #si #sec #sin #cos #ten #cot #math #nda #cds #pi/2 #180 #90 #1200 #classy #gdlmx #gd

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devphilamaths · 3 years

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Solve cosec(pi/2+theta) | cosec pi/2 + x formula, csc (pi/2 +x) | Find value cosec pi by 2 + x

#Solve cosec(pi/2+theta)#cosec pi/2 + x formula #csc (pi/2 +x)#Find value cosec pi by 2 + x #math #12th class #formula #sin #cos #ten #cosec #sec #cot

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hoctracnghiem · 3 years

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Giải phương trình $\cot x = – 1$.

Câu hỏi: Giải phương trình $\cot x = – 1$. A. $x = – \dfrac{\pi }{2} + k\pi \,\,\left( {k \in \mathbb{Z}} \right)$ B. $x = – \dfrac{\pi }{4} + k\pi \,\,\left( {k \in \mathbb{Z}} \right)$ C. $x = \pi + k\pi \,\,\left( {k \in \mathbb{Z}} \right)$ D. \(x = – […] from Học trắc nghiệm Hoc Trac nghiem

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mathematicianadda · 4 years

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Value of $\sum_{n=1}^{\infty} \dfrac{\cos (n)}{n}$ https://ift.tt/eA8V8J

I was trying to calculate the value of the series $\displaystyle \sum_{n=1}^{\infty} \dfrac{\cos (n)}{n}$ and I got an answer which I think could be right, but I'm not sure about some of the steps I took to get there. I was wondering if someone could provide some more insight so I can clear my doubts, and also check if I actually got the correct value.

First of all, I used Dirichlet's test for the convergence of the series, since $a_n = \dfrac{1}{n}$ is monotonic and $\displaystyle \lim_{n \to \infty} a_n = 0$, and the cosine partial sums can be bounded by a constant not dependent on $n$ (I'm pretty sure this is right since I looked other ways to do it, so I won't list exactly what I did to get the bound).

With that out of the way, I tried taking the expression $\dfrac{\cos(n)}{n}$ and rewriting it as something I could attempt to sum, and got this:

$$\displaystyle \int_1^{\pi} \sin(nx) \, dx = \left. -\dfrac{\cos(nx)}{n} \right|_1^{\pi} = \dfrac{(-1)^{n+1}}{n} + \dfrac{\cos(n)}{n}$$

$$\displaystyle \int_1^{\pi} \sin(nx) \, dx + \dfrac{(-1)^{n}}{n} = \dfrac{\cos(n)}{n}$$

And then

$$\displaystyle \lim_{n \to \infty} \displaystyle \sum_{k=1}^{n}\left(\displaystyle \int_1^{\pi} \sin(kx) \, dx + \dfrac{(-1)^{k}}{k}\right) = \displaystyle \lim_{n \to \infty} \displaystyle \sum_{k=1}^{n} \dfrac{\cos(k)}{k}$$

Then I tried separating the left side member into two sums, since

$$\displaystyle \sum_{n=1}^{\infty} \dfrac{(-1)^n}{n} = \displaystyle -\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{n} = -\ln (2)$$

I believe the latter equality can be derived using the alternate series test for the convergence of the series, and the Taylor expansion around $x = 0$ of $\ln {(1+x)}$ along with Abel's theorem. As for the other sum, this is the step I'm not sure about. I did

$$\displaystyle \lim_{n \to \infty} \displaystyle \sum_{k=1}^{n}\left(\displaystyle \int_1^{\pi} \sin(kx) \, dx\right) = \displaystyle \lim_{n \to \infty} \displaystyle \int_1^{\pi} \left(\displaystyle \sum_{k=1}^{n} \sin(kx)\right) \, dx$$

I'm not sure that's valid, and if it is I'm not sure why: I thought it would be fine since the partial sums could be arranged that way before taking the limit, but I suspect this thinking isn't correct, and I can't just swap the sum and the integral anytime without affecting the result. But anyways, if we take it as valid, then we can get a value for the sum by doing

$$\cos {(nx+\dfrac{x}{2})} - \cos {(nx-\dfrac{x}{2})} = -2\sin {(nx)}\sin{\left(\dfrac{x}{2}\right)}$$

$$\sin{(nx)} = \dfrac{\cos {(nx-\frac{x}{2})} + \cos {(nx+\frac{x}{2})}}{2\sin{\left(\frac{x}{2}\right)}}$$

And then

$$\displaystyle \sum_{k=1}^{n} \sin{(kx)} = \displaystyle \sum_{k=1}^{n} \dfrac{\cos {(kx-\frac{x}{2})} + \cos {(kx+\frac{x}{2})}}{2\sin{\left(\frac{x}{2}\right)}}$$

Which telescopes to

$$\displaystyle \sum_{k=1}^{n} \sin{(kx)} = \dfrac{\cos {\left(\frac{x}{2}\right)}-\cos {\left(\frac{2n+1}{2} \cdot x\right)}}{2\sin{\left(\frac{x}{2}\right)}}$$

Returning to the integral, we need to evaluate

$$\displaystyle \lim_{n \to \infty} \displaystyle \int_1^{\pi} \left(\displaystyle \sum_{k=1}^{n} \sin(kx)\right) \, dx = \displaystyle \lim_{n \to \infty} \displaystyle \int_1^{\pi} \frac{\cos {\left(\frac{x}{2}\right)}-\cos {\left(\frac{2n+1}{2} \cdot x\right)}}{2\sin{\left(\frac{x}{2}\right)}} \, dx$$

I again tried separating it in the sum of the integrals. The first one

$$\displaystyle \int_1^{\pi} \frac{\cos {\left(\frac{x}{2}\right)}}{2\sin{\left(\frac{x}{2}\right)}} \, dx = \displaystyle \int_{\sin {\frac{1}{2}}}^1 \dfrac{1}{u} \, du = -\ln({\sin{\frac {1}{2}}})$$

Via substitution $u = \sin{\frac{x}{2}}$

This won't change when $n$ goes to infinity. As for the second one

$$-\dfrac{1}{2} \displaystyle \int_1^{\pi} \dfrac{\cos{\left(nx+\frac{x}{2}\right)}}{\sin{\left(\frac{x}{2}\right)}} \, dx = -\dfrac{1}{2}\left(\displaystyle \int_1^{\pi} \dfrac{\cos{(nx)}\cos{\left(\frac{x}{2}\right)}}{\sin{\left(\frac{x}{2}\right)}} \, dx - \displaystyle \int_1^{\pi} \sin(nx) \, dx \right) = $$

$$= -\dfrac{1}{2}\left(\displaystyle \int_1^{\pi} \dfrac{\cos{(nx)}\cos{\left(\frac{x}{2}\right)}}{\sin{\left(\frac{x}{2}\right)}} \, dx + \displaystyle \left. \frac{\cos(nx)}{n} \right|_1^{\pi} \right)$$

Both of these integrals go to 0 as $n$ goes to infinity, applying the Riemann-Lebesgue lemma for the first one, since the function $f(x) = \cot{\left(\frac{x}{2}\right)}$ is continuous on $[1,\pi]$. Putting it all together gives

$$\displaystyle \displaystyle \sum_{n=1}^{\infty} \dfrac{\cos(n)}{n} = -\ln2-\ln{\left(\sin{\frac{1}{2}}\right)} = \boxed{-\ln{\left(2 \cdot \sin{\frac{1}{2}}\right)}} \approx 0.0420195$$

I used Octave to try and check the result: setting $n = 10^6$ gave me

$$S_{10^6} \approx 0.042020$$

Because of this, I'm inclined to think I got the correct answer, but I still doubt some of the steps I took (mainly the interchanging sum and integral one).

Thanks in advance. I'm sorry if I didn't make myself clear, english isn't my first tongue. I did some search as to find something related to this value, but couldn't find anything. Very sorry if its been answered before.

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