#range of inverse trigonometric functions

[inversetrigonometricfunctions]

Class 12 Maths Inverse Trigonometric Functions Course | JEE Mains की तैयारी | MathYug

अगर आप Class 12 Maths में Inverse Trigonometric Functions अच्छे से समझना चाहते हैं और साथ ही JEE Mains की भी तैयारी कर रहे हैं, तो MathYug का यह कोर्स आपके लिए बहुत फायदेमंद है। इस कोर्स में expert teacher Ashish Sir ने हर concept को आसान भाषा में समझाया है।

आप इस course में सीखेंगे:

Inverse Trigonometric Functions की definition

Domain और range

Principal value branches

Graphs और identities

Elementary properties

सिर्फ समझना ही नहीं, बल्कि concept को और पक्का करने के लिए NCERT textbook, exemplar, RD Sharma और Board’s Question Bank से चुने गए questions को assignments के तौर पर दिया गया है। इन assignments की PDF भी download की जा सकती है।

यह कोर्स Class 12 students के साथ-साथ JEE Mains aspirants के लिए भी designed है, ताकि दोनों ही exams की तैयारी एक साथ हो सके।

MathYug आपको expert guidance, professional notes, और structured learning के साथ एक complete learning experience देता है। Join कीजिए और अपने maths concepts को strong बनाइए।

Join the complete course now at: https://mathyug.com/c/ch2-inverse-trigonometric-functions/

[sethiswiftlearn]

Class 12 Math | Mastering Inverse Trigonometric Functions | JEE Mains, CUET Preparation

Inverse Trigonometric Functions play a crucial role in Class 12 Mathematics, forming the foundation for calculus and higher-level mathematics. A strong grasp of these concepts is essential for excelling in JEE Mains, CUET, and other competitive exams. This session is designed to help students master the properties, domain, range, and applications of inverse trigonometric functions with easy explanations and problem-solving techniques.

What You’ll Learn in This Session?

✔ Definition & Basic Concepts – Understanding inverse trigonometric functions and their significance. ✔ Domain and Range – A clear breakdown of how to determine valid input values. ✔ Principal Branches & Graphs – How to visualize and interpret inverse trigonometric functions. ✔ Important Identities & Formulas – Mastering standard formulas for solving complex problems. ✔ Differentiation & Integration of Inverse Trigonometric Functions – A critical aspect for calculus-based questions. ✔ Exam-Focused Problem Solving – Previous year JEE Mains, CUET, and CBSE Board questions with step-by-step solutions.

Why Watch This Session?

Boost Conceptual Understanding – Strengthen your fundamentals for competitive exams.

Solve Problems Faster – Learn shortcuts and tricks to tackle tricky questions easily.

High-Weightage Topic – This topic frequently appears in JEE Mains, CUET, and Board exams.

Step-by-Step Explanations – Detailed solutions to help you gain confidence in solving problems.

📌 Watch Now & Ace Your Math Exams! 👉 https://youtu.be/Y1o0AlflWsI?si=A-VnLoDg6vNwlUrl

🔔 Subscribe for More! Stay tuned for more revision sessions, exam tricks, and concept explanations to help you master Class 12 Mathematics. 🚀

[surajkumasblog]

JavaScript Math.asin() - Arcsin Function Explained

The javascript arcsin computes the arcsine (inverse sine) of a number. The result is returned in radians, ranging between -π/2 and π/2.

Syntax:

Math.asin(x)

Parameters:

x: A number between -1 and 1.

Return Value:

Returns the arcsine of x in radians. If x is outside the range [-1, 1], it returns NaN.

Example Usage:

console.log(Math.asin(1)); // Output: 1.5707963267948966 (π/2) console.log(Math.asin(0)); // Output: 0 console.log(Math.asin(-1)); // Output: -1.5707963267948966 (-π/2) console.log(Math.asin(2)); // Output: NaN (out of range)

To convert radians to degrees:

function toDegrees(radians) { return radians * (180 / Math.PI); } console.log(toDegrees(Math.asin(0.5))); // Output: 30 degrees

This function is widely used in trigonometric calculations.

[subhasapco123]

Understanding Inverse Trigonometric Functions: A Comprehensive Guide

Trigonometry is one of the foundational subjects in mathematics that finds applications in various fields such as physics, engineering, and even computer science. While trigonometric functions like sine, cosine, and tangent describe relationships between the sides and angles of a right triangle, inverse trigonometric functions are equally essential for solving problems that involve angles when the sides of the triangle are known.

Inverse trigonometric functions, as the name suggests, are the reverse of the standard trigonometric functions. This blog will explore the concept of inverse trigonometric functions, their properties, and how they are used in mathematical and real-world applications.

What are Inverse Trigonometric Functions?

The inverse trigonometric functions are the functions that reverse the action of the regular trigonometric functions. In simple terms, while a regular trigonometric function takes an angle and gives a ratio of sides (such as sine giving opposite/hypotenuse), an inverse trigonometric function takes a ratio and gives an angle.

The six trigonometric functions in mathematics are:

Sine (sin)

Cosine (cos)

Tangent (tan)

Cotangent (cot)

Secant (sec)

Cosecant (csc)

Each of these functions has an associated inverse function. For example, the inverse of sine is called arcsine (or sin⁻¹), the inverse of cosine is called arccosine (cos⁻¹), and so on.

Why Do We Need Inverse Trigonometric Functions?

Inverse trigonometric functions are crucial because they allow us to find the angle when we know the value of the trigonometric function. This is particularly useful in fields like navigation, physics, engineering, and computer graphics, where it’s essential to work backward from a ratio of sides to determine the angle.

For instance, if we know the sine of an angle in a right triangle, the inverse sine (sin⁻¹) function can help us determine the measure of the angle. Similarly, inverse functions like arctangent (tan⁻¹) help us find the angle when the ratio of the opposite side to the adjacent side is known.

The Notation of Inverse Trigonometric Functions

The notation for inverse trigonometric functions is a bit different from regular trigonometric functions. Instead of writing "sin(x)" or "cos(x)," the inverse trigonometric functions are denoted with a superscript minus one, such as sin⁻¹(x) or cos⁻¹(x). This notation represents the angle whose sine or cosine is the given value.

Here’s a quick list of the common inverse trigonometric functions:

sin⁻¹(x) or arcsin(x): The inverse of sine, gives the angle whose sine is x.

cos⁻¹(x) or arccos(x): The inverse of cosine, gives the angle whose cosine is x.

tan⁻¹(x) or arctan(x): The inverse of tangent, gives the angle whose tangent is x.

cot⁻¹(x) or arccot(x): The inverse of cotangent, gives the angle whose cotangent is x.

sec⁻¹(x) or arcsec(x): The inverse of secant, gives the angle whose secant is x.

csc⁻¹(x) or arccsc(x): The inverse of cosecant, gives the angle whose cosecant is x.

Domains and Ranges of Inverse Trigonometric Functions

One of the critical aspects of inverse trigonometric functions is that they are restricted to certain domains and ranges to ensure that they are one-to-one functions. A one-to-one function is essential because it ensures that each input corresponds to a unique output.

Arcsin (sin⁻¹):

Domain: -1 ≤ x ≤ 1

Range: -π/2 ≤ y ≤ π/2

The arcsin function gives an angle between -90° and 90°.

Arccos (cos⁻¹):

Domain: -1 ≤ x ≤ 1

Range: 0 ≤ y ≤ π

The arccos function gives an angle between 0° and 180°.

Arctan (tan⁻¹):

Domain: -∞ < x < ∞

Range: -π/2 < y < π/2

The arctan function gives an angle between -90° and 90°.

Arccot (cot⁻¹):

Domain: -∞ < x < ∞

Range: 0 < y < π

The arccot function gives an angle between 0° and 180°.

Arcsec (sec⁻¹):

Domain: |x| ≥ 1

Range: 0 ≤ y ≤ π/2 or π ≤ y ≤ 3π/2

The arcsec function gives an angle between 0° and 90° or between 90° and 180°.

Arccsc (csc⁻¹):

Domain: |x| ≥ 1

Range: -π/2 ≤ y ≤ 0 or 0 ≤ y ≤ π/2

The arccsc function gives an angle between -90° and 90°, excluding 0°.

Properties of Inverse Trigonometric Functions

Understanding the properties of inverse trigonometric functions can make working with them much easier. Here are some essential properties:

Inverse of an Inverse: The inverse of an inverse trigonometric function gives the original function back. For example:

sin(sin⁻¹(x)) = x for -1 ≤ x ≤ 1

cos(cos⁻¹(x)) = x for -1 ≤ x ≤ 1

tan(tan⁻¹(x)) = x for all x

Composition of Functions: The inverse and the original trigonometric function can be composed together. For example:

sin⁻¹(sin(x)) = x for -π/2 ≤ x ≤ π/2

cos⁻¹(cos(x)) = x for 0 ≤ x ≤ π

tan⁻¹(tan(x)) = x for -π/2 < x < π/2

Symmetry: Inverse trigonometric functions exhibit symmetry about certain axes. For example, the inverse sine function is symmetric about the y-axis, while the inverse cosine function is symmetric about the line x = 0.

Solving Trigonometric Equations Using Inverse Functions

Inverse trigonometric functions are widely used for solving trigonometric equations. For example, if you are given a problem where you need to find the angle θ, knowing the value of sin(θ) = 0.5, you can use the arcsin function to find the angle:θ=sin⁡−1(0.5)=30∘θ = \sin^{-1}(0.5) = 30^\circθ=sin−1(0.5)=30∘

Similarly, if you are given the tangent value of an angle, you can use the arctan function to find the angle. This process is vital for solving problems in geometry, calculus, and physics.

Real-World Applications of Inverse Trigonometric Functions

Navigation: Inverse trigonometric functions are crucial in navigation and determining bearings. Pilots and sailors use these functions to calculate angles based on given distances and directions.

Physics: In physics, especially in wave motion and optics, inverse trigonometric functions help solve problems involving angles of refraction, angles of incidence, and angular displacement.

Engineering: In electrical engineering and mechanical systems, inverse trigonometric functions are used in control systems, signal processing, and analyzing vibrations.

Computer Graphics: Inverse trigonometric functions are used in computer graphics to rotate and scale objects, especially when working with angles in 3D space.

Conclusion

Inverse trigonometric functions are indispensable tools for solving mathematical and real-world problems involving angles and ratios. From geometry to physics and engineering, they provide a method for determining the angle when the side ratios of a right triangle are known. Understanding the properties and applications of inverse trigonometric functions will undoubtedly help you excel in both theoretical and applied mathematics.

[clikere]

Simplify Your Math Problems with Calculator

Describe a calculator

A calculator is a tool used to compute mathematical operations on numbers. Simple calculators can only perform the mathematical operations of addition, subtraction, multiplication, and division. Exponential operations, square roots, logarithms, trigonometric functions, and hyperbolic functions can all be handled by more advanced calculators. Some calculators internally carry out all these operations through a series of additional operations.

The development of calculators

Nowadays, the majority of calculators are either battery-powered or powered by electricity. Calculators operate by carrying out preprogrammed operations using numerical inputs. The slide rule, a more basic calculator, was widely used before the invention of the electronic calculator (about 1970). It was made of a slide, a wooden slat that could be inserted and removed from a pair of slats that had been reinforced. There were calibrated numerical scales on both the slide and the outside pair of slats.

A simple calculator is exemplified

The cursor, a moveable, translucent sleeve, was used to align the scales' numbers. Although there was no need for a power source, the slide rule's precision was limited, and mastering it required a learning curve. In some parts of the Far East, one of the oldest calculators—the abacus—is still in use. The abacus functions as an adding machine that represents numbers with groups of beads. The abacus does not require a power source, like the slide rule. The beads can be moved up and down to represent arithmetic operations and are arranged in numerous parallel rows. According to legend, an adept abacus operator may perform some computations just as fast as a person equipped with a battery-powered calculator.

Contemporary calculators

Calculators were able to perform calculations for addressing problems involving variables (unknowns) as they developed during the 1970s. The first personal computers were these. Additionally, digital calculators are available as downloads or are pre-installed on the majority of smartphone and personal computer operating systems. Modern personal computers are still capable of carrying out these tasks, and most of them come with a virtual calculator programme that simulates a handheld calculator on screen. By pointing and clicking, the buttons are activated.

Theoretically, a modern computer is a calculator with a considerably greater memory that uses binary numbers. The vast array of non computational functions that a computer is capable of, however, make it far more than just a simple calculator in the real world.

Various calculator types

Over the past several decades, calculators have undergone changes. Today, a wide range of calculators from manufacturers like Casio and Texas Instruments are accessible to meet various mathematical needs. Let's examine some of them.

Simple calculators

The four-function calculator, which can perform fundamental mathematics including addition, subtraction, multiplication, and division, is the most basic type of calculator. Four-function calculators can produce decimal values and typically have a +, -, x, and / sign to signify the operations. Additionally, some contain a% buttons that are used to compute percentages.

Mathematical calculators

The scientific calculator is intended for use in completing scientific calculations, as its name suggests. Since it must be able to do trigonometric functions, logarithms, sine/cosine, and exponential operations, this sort of calculator typically includes more buttons than a conventional calculator. The display on a scientific calculator is typically bigger so you can see longer equations and more digits at once.

Conclusion

It is possible to use a scientific calculator to solve math, science, and engineering problems. It has specialized buttons for parentheses, exponents, inverses of trigonometric functions, trigonometric functions, and. There are distinct rules for each scientific calculator. Go to www.clickere.in to get a scientific calculator. 

Tags: Stationery, Writing & Correction and Scientific Calculator.

[manasastuff-blog]

Introduction

Welcome to Mansa Defence Academy, where we provide the best coaching to students aspiring to join the National Defence Academy (NDA). This blog post aims to provide you with a comprehensive understanding of the Group C syllabus for the NDA examination. Whether you are a first-time candidate or looking to refresh your knowledge, this blog post will equip you with the necessary information to excel in the examination.

Section 1: Mathematics

Mathematics is an integral part of the NDA Group C syllabus. It covers various such as algebra, trigonometry, calculus, and. Mastering these topics is crucial to scoring well in the examination. At Mansa Defence Academy, we have experienced faculty members who provide in-depth coaching on each of these areas.

Algebra

Algebra forms the foundation of mathematical concepts in the NDA examination. It includes topics such as equations, polynomials, and functions. Our faculty members break down complex algebraic concepts into easy-to-understand modules, ensuring that you grasp the fundamentals effectively.

Trigonometry

Trigonometry is another essential topic in the mathematics section of the NDA Group C syllabus. It covers concepts like trigonometric ratios, identities, and inverse trigonometric functions. Our faculty members simplify these concepts by relating them to real-life scenarios, making it easier for you to remember and apply them during the examination.

Calculus

Calculus involves the study of limits, derivatives, and integrals. While it may seem daunting at first, our experienced faculty at Mansa Defence Academy utilize interactive teaching methods to make calculus concepts more intuitive and accessible. By providing real-life examples and applications of calculus, we ensure that you develop a strong understanding of this topic.

Statistics

Statistics plays a significant role in the NDA examination. It includes topics such as probability, measures of central tendency, and correlation. At Mansa Defence Academy, we believe in promoting a practical approach to statistics. Our faculty members emphasize the application of statistical concepts in military scenarios, enabling you to grasp the subject matter effectively.

Section 2: English

The English section of the NDA Group C syllabus assesses candidates' proficiency in grammar, vocabulary, and comprehension. It is essential to have a strong command of the English language to perform well in this section.

Grammar

Grammar forms the core of the English language. It includes topics such as tenses, parts of speech, and sentence structure. Our experienced faculty members at Mansa Defence Academy focus on breaking down complex grammar rules into simplified modules. We also provide practice exercises to help you reinforce your understanding of these concepts.

Vocabulary

A wide-ranging vocabulary enhances your ability to understand and communicate effectively. The NDA examination tests candidates' vocabulary through synonyms, antonyms, and idiomatic expressions. At Mansa Defence Academy, we offer vocabulary-building exercises, word lists, and context-based learning to help you expand your vocabulary repertoire.

Comprehension

Comprehension passages test your ability to understand and derive meaning from written texts. Our faculty members guide you through various comprehension strategies, such as skimming, scanning, and inference, enabling you to comprehend passages efficiently. Regular practice with different types of passages further enhances your reading comprehension skills.

Conclusion

Navigating through the NDA Group C syllabus can be a challenging task. However, with the right guidance and coaching, you can excel in the examination. At Mansa Defence Academy, we are committed to providing you with the best coaching experience, ensuring that you are well-prepared for the NDA examination. Enroll now and embark on your journey towards a successful career in the defense forces.

[kiskisur]

Inverse Trigonometric Functions:

Because the six main trigonometric functions are periodic, they are not injective (or, 1 to 1), and thus are not invertible. By restricting the domain of a trigonometric function, however, they can be made invertible.

The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:

took off my glasses and stepped on it never reading again

[thepenpostblog]

Trigonometry Formulas

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It’s a fundamental topic in mathematics and finds applications in various fields, from physics and engineering to astronomy and navigation. To master trigonometry, one needs to be well-versed in its numerous formulas and identities. In this comprehensive guide, we will explore a wide range of trigonometry formulas, making them accessible and understandable. Whether you’re a student looking to ace your trigonometry exams or a professional seeking to apply trigonometric principles in real-life scenarios, this article has got you covered.

Trigonometry Formulas List:

Let’s kick things off by presenting a list of the fundamental trigonometry formulas that we will delve into in more detail:

Basic Trigonometric Function Formulas

Reciprocal Identities

Trigonometry Table

Periodicity Identities (in Radians)

Cofunction Identities (in Degrees)

Sum & Difference Identities

Double Angle Identities

Triple Angle Identities

Half Angle Identities

Product Identities

Sum to Product Identities

Inverse Trigonometry Formulas

Trigonometry Formulas from Class 10 to Class 12

Trigonometry Formulas in Major Systems

[calebmatthew1]

The Ultimate Triangle Companion: Maximizing Efficiency with the Triangle Calculator

A user-friendly allcalculator.net’s Triangle Calculator helps users ascertain a Triangle's perimeter and area in seconds. It has a user interface that assists people to get to know about finding an angle and side of a Triangle. 

Whether you’ve given three sides, two sides, or just an angle or two angles, our triangle side calculator helps to solve every geometry problem. The triangle angle calculator helps to determine the angles and sides of the triangle, whether it is an acute, obtuse, equilateral, isosceles, or right triangle. 

You can try our area of a triangle calculator to help you find out the area and perimeter instantly. It measures the three sides and angles, providing error-free results in just a single click. 

The Triangle calculator online facilitates you to estimate the area needed to form a triangle figure. Though it covers the most common usages, it includes an included angle and two sides to find the missing triangle side and length. It has lucrative benefits to apply in our daily lives. 

How Efficient Is A Handy allcalculator.net’s Triangle Calculator? 

Our area of a triangle calculator is a user-centric tool that facilitates individuals to solve problems consisting of the angles and lengths of right triangles. It helps the students to build reasoning and critical thinking ability confidently and effectively. Let’s sum up a couple of strategies through which our area of triangle formula can help you:

Beneficial in explaining the Triangular Basics: Before you know the concept of our triangle calculator, ensure that you understand the terminology and concepts of right triangles. Go through the triangular legs, Pythagorean theorem, and hypotenuse, and then discover the adjacent, hypotenuse, and opposite sides of a triangular angle. 

Using triangle side calculator, you can distinguish between radians and degrees and know how to convert them easily. Try real-life examples, exercises, and diagrams to motivate the triangular concepts. 

Use allcalculator.net’s Triangle Calculator to check the Errors: Once the students know the concepts and basics about the triangle calculator angle, it can show them the right way to use the online triangle calculator. 

You’ll need to understand the functions and features of the triangle calculator right angle to enter accurate values, reset the screen, switch the modes and check the probable mistakes. Using our side of triangle calculator, you can understand the tricks and shortcuts to use it for the inverse functions and fractions and recall and store the known values. 

Use our Triangle Calculator with Real-life Examples: Once you optimize the triangle calculator, you can allow the students to practice real-life problems with our area of triangle formula. To generate your own, you can try problems with worksheets, textbooks, or online sources. 

Ensure the problems cover a wide range of scenarios and conditions like finding the missing angles, missing sides, trigonometric rations, and finding the perimeters. With solve a triangle calculator, you can understand real-world applications and problems to convert your practice more engaging and relevant. 

Helps to motivate Exploration into the Triangle problems: When your practice problems with allcalculator.net’s Triangle calculator, it can encourage you to discover various ways to solve the different or same problems using the same values. You can use this online tool to determine the sides and angles of a right triangle with the same hypotenuse. 

Using our triangular prism volume calculator, you can challenge the answers to find the right relationships, formulas, or patterns involving the right triangles with an online calculator. It helps to develop creativity, problem-solving, and critical thinking abilities. 

Helps to offer you Accurate Feedback: The most vital part of using our side of triangle calculator is to offer the users accurate feedback. It helps individuals to enhance their accuracy by correcting their mistakes, thereby increasing their overall confidence. 

With our triangle area calculator, you can access the feedback in various ways by verifying answers, lifting the efforts, providing them hints, and asking the required questions. You can try peer feedback, under which learners can get constructive comments. The feedback needs to be specific, positive, and timely. 

[ashishkumarletslearn]

#InverseTrigonometricFunctions #Class12Maths

[mathsodology]

23. Compositions of trigonometric and inverse trigonometric functions

23. Compositions of trigonometric and inverse trigonometric functions

Proof. (1) Click here (2) Click here (3) Click here (4) Click here (5) Click here (6) Click here (7) Click here (8) Click here (9) Click here (10) Click here (11) Click here (12) Click here (13) Click here (14) Click here (15) Click here (16) Click here (17) Click here (18) Click here (19) Click here (20) Click here (21) Click here (22) Click here (23) Click here (24) Click…

[sethiswiftlearn]

📚 Class 12 Math | Inverse Trigonometric Functions | Domain, Range & Applications | JEE Mains, CUET Prep 🎯

Master Inverse Trigonometric Functions with this in-depth session! Understand the domain, range, and key applications of inverse trig functions, a crucial topic for JEE Mains, CUET, and board exams. Learn problem-solving techniques, important properties, and real-world applications to boost your math skills and exam performance! 🚀

📌 What You’ll Learn: ✅ Domain & Range of Inverse Trigonometric Functions ✅ Important Properties & Identities ✅ Graphs & Transformations ✅ Application-based Problems for JEE & CUET

👉 Watch Now: https://youtu.be/XMNxJlvHYBM?si=M75xA737zc3e28iQ

🔔 Don’t forget to Like, Share & Subscribe for more expert math lessons!

[cherrysung]

attention

pairing: nerd!jaemin x reader

genre: smut

warnings: language, unprotected sex (stay safe!), deepthroating, oral (m. receiving), creampie, choking, fingering, dirty talk, slight degradation

prompts: none

summary: innocent and harmless to the eyes of everyone else, the quiet boy sitting next to you couldn’t stand your impatience anymore. maybe another lesson needed to be taught.

requested by @jjuullnnaayyaa.

word count: 2.5k

note: I hope you like this! this is the second part (requested) to my first nerd!jaemin scenario, which you can check out here! also, @glossyjaems have fun lmao

cherrysung’s navigation

“Stop it.”

“Stop what? I’m not doing anything.”

Clenched jaw and threatening, narrowed eyes, Jaemin could only spare your mischievous smile a short glance before turning his attention back to the boring teacher in front of you. She spoke about some physics formulas and confusing projectile motions, yet although your focus seemed to be directed with full interest at her explanations, it was far from that. Instead, while you stared at the white board covered in multiple doodles, your hand rested not so innocently on Jaemin’s thigh. All under the desk and for nobody else to see.

You scribbled down the long formulas on your notebook in an unbothered nature, left hand moving slowly in up and down motions on your desk partner’s thigh, occasionally squeezing and teasing to reach unholy areas. Jaemin appeared to be sucking in ragged breaths, the spectacles sitting on the bridge of his nose slightly sliding down due to the increasing perspiration on his face.

He looked nervous, angered even. Gritting his teeth or biting on his lower lip as your hand gradually moved higher and closer to the growing bulge inside his beige dress pants.

The two of you had been seeing each other for a considerable amount of weeks now, thankful that a simple tutoring session was enough to kick off your current relationship; date after date, you always picked up a new characteristic from his reserved personality, only adding to your belief that really—not all is what it appears to be. Despite the fact Jaemin was in reality a freak and nothing like the nerd he had come to be recognized as, he was still fairly shy, and had struggled big time to ask you to be his girlfriend. Stumbles and stutters had been leaving his lips infinitely until you kissed him and spared him the work, answering his timid question with a simple yes.

His breath abruptly hitched, hands roughly gripping onto the desk’s edge and pencil when you curiously palmed his hardened cock. The muscles on his thighs were flexed, Adam’s Apple bobbing up and down as he forcefully swallowed with nervousness and want.

“Stop that. You know what I’m talking about.” He whispered, eyes still fixated on the class before the two of you whilst using his free hand to grip yours gently and stop it from further movements. “What’s with you? We’re in the middle of class, we need to pay attention or we won’t understand anything afterwards.”

You smiled, admiring his control and obedience. Leaning closer to his ear, you whispered anyways, “but the only one I want to pay attention to is you. Jaemin, I want to fuck you.”

In a bold move, you guided his hand under your plaid skirt and towards your soaked heat, moving your flimsy underwear to the side to run his index finger through your wet folds.

Jaemin cursed under his breath, tongue swiping swiftly on his bottom lip as he lowered his head so the teacher wouldn’t notice the slight pink tint that had flushed warmth into his cheeks. His fingers started moving on their own, thumb rubbing your clit gently before plunging two fingers into your dripping hole. You allowed him to continue the discrete assault on your pussy as you brought your hand back to its previous position—his dick. Experimentally, you unbuttoned his pants with only one hand, taking his hard member out after he had obediently raised his hips to lower down the fabric of his black underwear.

He continued to pump his fingers in and out of your cunt at an average pace, unable to move them any faster due to the little space between you and the desk. Still, he didn’t fail to please you, curling them and hitting your sensitive spot successfully with no trouble whatsoever. Meanwhile, you stroked his length slowly and tightly, using the precum leaking from the tip to coat his dick. All of this while he innocently rested his chin on his hand, mindlessly nodding at whatever the teacher said and as you pretended to listen, now writing random things on your notebook, because you weren’t bothered enough to actually keep scribbling down notes from the subject.

“So, Na Jaemin, what do we do next? Give a brief explanation too.” The teacher spoke up, all heads turning your way and prompting both your movements to halt for a moment. Though, it didn’t take long for them to resume, as if twenty-eight eyes weren’t situated on the boy next to you. If only they knew what was happening underneath the metal table.

Jaemin fixed his glasses, and you could only widen your eyes in annoyance at how fast his demeanor changed, gaze turning shy and cheeks still warm from your ministrations; but, fingers reaching deeper while his thumb returned to caressing your sensitive bub. “You move the number three to the other side of the formula,” he spoke with confidence, hips involuntarily jolting up slightly when your thumb swiped for a little too long on his tip. “The reason why you have to do this, is to actually get the formula working. Otherwise, the answer will be unobtainable.”

“That is correct.” The teacher nodded with a small smile. “Thank you, Na Jaemin. I didn’t expect any less from my best student, anyways.”

For about ten more minutes, the lesson carried on at an extremely boring and deathly pace, the sound of markers coming into contact with the board and quiet complaints from students the only things that could be heard between the four, dull white walls. Jaemin and you had resorted to torturing each other in the form of teasing, unwilling to have any type of release underneath a dirty school desk.

The bell rang, signaling the end of the lesson as the two of you stopped your actions immediately. You swiftly fixed your skirt, and Jaemin made sure to zip his pants back up only after quickly licking his fingers clean from your essence while gazing deeply with desire into your eyes.

“Class, you’re dismissed.”

You stood up a ridiculously desperate pace, only to be pulled back down onto your seat by Jaemin. “What the fuck! Jaemin, I’m horny, let’s leave.” Your threats came out in a whisper, but they were enough to bring a sinister smile to Jaemin’s features.

“Don’t worry, I have a better idea.” He waited for the last pair of students to leave the classroom, watching as the teacher stood with her bag, hands holding onto her homemade lunch. She walked carelessly towards the door, steps coming to a halt upon noticing the two of you were still glued to your seats.

“Guys? It’s lunchtime, why are you still here?”

Jaemin took out his calculus textbook along with his calculator, placing the items on the table before pointing at them and then at you. “Could we stay here? She’s having trouble with Inverse Trigonometric Functions and we have calculus class after lunch break. We thought it’d be a good idea to stay here practicing just to warm up.”

She pondered for a few seconds, before nodding her head. “Sure, but if you get hungry don’t hesitate to go to the cafeteria, okay? It’s always better to avoid an empty stomach. Good luck and work hard!”

With that, she left, softly closing the door after praising the two of you for being such ‘dedicated students’. Jaemin chuckled quietly before locking the classroom door, walking back to the table and pulling you up rapidly, wasting no time in locking lips with you after having to put up with your endless teasing for an entirety of forty-five tedious minutes.

You held tightly onto his shoulders as he walked you backwards until your back hit a wall, taking advantage of the gasp that escaped your lips to kiss you deeper, tongue fighting with your own as his hands roamed with lust all throughout your body. He unbuttoned your blouse just enough to free your tits, pulling your bra down as his fingers pinched your nipples playfully. Jaemin swallowed your moans, turning the both of you around so his back was pressing against the wall now, rubbing your shoulders tenderly before pushing you down to your knees.

“You were so naughty today, weren’t you? I thought you had already gotten a well deserved lesson taught? Seems like you didn’t learn shit, my dirty princess. Should I be harder on you?”

A whimper left you as you kneeled on the cold, tiled flooring, looking up at Jaemin obediently as he lovingly tucked your hair behind your ears. He released his throbbing cock from his undergarments, right hand bringing your locks up into a makeshift ponytail before pushing your head closer to his leaking shaft.

“Do you have make-up removal wipes here?” He simply asked, free hand pumping his member as he looked down at you with hooded eyes.

You nodded.

“Then we’re fine. Open your pretty mouth, doll.”

You listened to his demand right away, mouth opening with your tongue slightly sticking out as Jaemin placed his cock on top of the warm muscle, breathing instantly becoming irregular. In a rapid motion that caught you off guard due to no warning given beforehand, he roughly thrusted his hips forward, dick reaching the back of your throat unexpectedly as you gagged, fighting the urge to cough out as his actions became consecutive and merciless.

He successfully caressed the depths of your throat experimentally, the outlines of his member occasionally showing whenever you were able to avoid the need to gag. His low grunts resonated throughout the empty classroom, fucking your mouth ruthlessly until tears mixed with black mascara were running down your face uncontrollably and your throat burned as it begged for an inexistent break.

His movements only slowed when his dick twitched, hand forcing you down onto his cock ‘till your nose was pressed against his tummy once last time before letting you go, gasps and coughs desperately leaving your lips in an instant as you attempted to catch your breath. Jaemin let out an exhale and pulled you back to your feet once again, swiftly wiping the black tears from your cheeks as he admired your reddened eyes. He wordlessly asked if you were fine, smiling gently when you nodded your head.

“Come here.” He guided you towards the teacher’s desk—again, giving you no time to process what was happening before he had pushed your upper body down, chest and cheek pressed onto the polished wood while his hand rested on your back. He pushed your skirt upwards, the fabric bunching up at your waist as he moved your pink panties to the side.

Your glimmering cunt stared back at his intense gaze, juices now running down your inner thighs after enduring the attack of his fingers with no room for release. He ran the tip of his dick up and down your folds teasingly, breath hitching while whines escaped you.

“Jaemin, please.” You whispered, mind unable to think of anything else but the feeling of his shaft caressing your pussy.

“What, princess? What do you want? Use your words.”

“Please, fuck me. Fuck me hard, so hard I won’t be able to focus the rest of the lessons. Fuck me so hard I will keep my hands to myself during class next time I feel like a little slut.”

“Good choice and wording, love. We only have about… ten minutes left before lunch break ends, so let’s make this worth it. Let me fuck the whore out of you, alright?”

No more words said, he rammed into you, setting a ferocious and reckless pace right away as your frail body lunged forward weakly with every fierce thrust that he delivered. The tip of his dick kissed at your sweet spot every time, eliciting gasps and whimpers from you that only served to help Jaemin increase his already inhumane speed.

“You like that right?” He gritted out. “This is what you wanted all along, right? You were so desperate for my dick that you couldn’t wait until we got home; no, you just had to touch me in the middle of a lesson. What a dirty doll you are.”

He continued to pounce into you, thrusts never decreasing in rhythm as he pulled you up close to his chest by your hair, free hand circling around your neck before squeezing lightly. You let out a particularly loud sound of pleasure, hands shooting up to rest on Jaemin’s much bigger one, silently begging him to keep the hold on your neck.

A surprised chuckle left his lips, hand squeezing once again as your walls clenched around him, tightly engulfing his dick deliciously. He moaned, head falling to your shoulder before biting gently onto the skin, sucking and then soothing the burning feeling with a slight swipe of his wet tongue. Whispering dirty things into your ear, you whined as his words swirled chaotically in your mind; promises of what more he’d do to you once you were alone and praises of how good you felt drowning you.

You felt his member twitch inside of you, thrusts gradually becoming sloppier by the moment. The grip on your neck became stronger, nearly preventing you of all ability to breath.

“J-Jaemin, I’m close!”

“Yes, princess, go ahead and let go. I want to feel you. Go ahead and cum all over my dick.”

Jaemin softly nibbled on your earlobe, the action enough to bring you to your climax. Your cunt clenched unstoppably around his shaft, juices mixing along with his as he reached his release as well, cum filling your walls to the brim through white, warm strings. His hips abruptly halted, dick still buried deep inside you while the two of you attempted to catch each other’s breaths.

Upon the loud sound of the bell ringing through the classroom and school halls, you both fixed yourselves in record time. After zipping up his pants, Jaemin helped you button up your shirt, fingers collecting his juices that had trickled down from your pussy before inserting the milky substance inside of you once again and arranging your underwear back in place, your legs shaking at the oversensitivity. He smirked and cupped your heat, eyes twinkling at the squeal that left your lips. “Keep that in you until we return back to my house.”

Quickly, he was able to unlock the door on time as the two of you now sat at your assigned desk, while Jaemin softly wiped off the black stains of mascara from your cheeks.

“Oh my God, what happened? Why is she crying?” The teacher asked worriedly the moment she stepped into the classroom.

“Oh, don’t worry,” Jaemin smiled sweetly at the elderly woman, “she became angry because she couldn’t get the topic, so angry that she started crying.”

“Oh, sweetie,” she smiled at you with pity, “you’ll get it soon! Don’t be too hard on yourself.”

“I know right! I told her the same thing.” Jaemin giggled, aware of the strong glare you sent his way. “Don’t worry, though, I promised her I will keep teaching her the lesson after school.” He smiled innocently, eliciting coos from the teacher who had been unaware of the slightly harsh pat that he delivered to your sensitive heat.

You gulped, rubbing your thighs together at the feeling of his cum still sitting in the depths of your walls, and the intrigue of what else he would ‘teach’ you behind closed doors.

[mocktopper]

Class 12 Maths Chapter 2 Inverse Trigonometric Functions

Class 12 Maths Chapter 2 Inverse Trigonometric Functions

Question 1: Find the principal value of Answer: Let sin-1 Then sin y = We know that the range of the principal value branch of sin−1 is and sin Therefore, the principal value of Question 2: Find the principal value of Answer: We know that the range of the principal value branch of cos−1 is . Therefore, the principal value of. Question 3: Find the principal value of cosec−1 (2) Answer: Let…

[noctumsolis]

On quantiles

Coding is a balancing act. When it comes to quantiles for non uniform RNG the balance is between accuracy and execution speed. My focus has been on achieving a standard normal distribution RNG.

Because of the goal of the project, I'm doing in a this in a C++ environment where I can not use C++11 or the boost libraries. As a result I must implement a solution myself. At first I used the Box Muller transform, but its an unwieldy beast. It employs both square root and log functions and requires dual random values (which isn't so bad as it produces two values as well). Its output is not ideal as it doesn't handle symmetry well and produces data ranges with peculiar holes. All in all it's not great.

There's another popular algorithm for large sets which is the Ziggurat algorithm. It's good, but I have reasons for not using it that largely boil down to the fact that one of my goals is to create an easy-to-use library; Ziggurat is not easy to use.

At this point in my reading I encountered probit; a quantile function for the normal distribution that works well with semi-random numbers. The equation for it is pretty straight forward, except.

Except that it relies on an inverse error function. Can that be implemented? Sure. But this is where I get in my own way; erfinv is ugly. Every implementation I've seen uses tables and long equations mediated by multiple conditionals. It's not inefficient, it's not unreasonably imprecise, it's just ugly. I don't like writing ugly code in my personal projects.

But probit has an interesting counterpart; logit is the equivalent quantile for the logistic distribution. Overlay logistic and normal curves and you can see a lot of similarities. This, I felt, could suffice if needs must. Best of all, it's a graceful little function. It does use a log, but only one. It's gorgeous!

But it's not right, really. Trying to pass off logit as though it gave you a normal distribution. I needed a solution. It needn't be perfect, just good enough for the target use (which is in games). So I resolved to modify logit, to introduce what I think might be called a coefficient.

I started with a plot of probit, logit, and the different between them. Then I stated tinkering, trying to match another curve - with a simple equation - to that product. What I've achieved so far isn't perfect, but it's probably close enough for my purpose and I think that it could be tweaked further, with minimal added complexity.

It does introduce a trigonometric function to the mix, but it feels graceful. Best of all it produces output that, in my opinion, is superior to the Box Muller algorithm. It's more naturally balanced and doesn't seem to have the troubling gaps.

I'll probably find time to compare the computational efficiency and might yet try for a better fit to the target curve, but for now I'm happy.

And I'm calling it fauxbit.

[ashishkumarletslearn]

#Class12Maths #InverseTrigonometricFunctions

Get complete video lecture, pdf notes and assignments for Cbse Chapter 2 Ranges of Inverse Trigonometric Functions Class 12 Maths.