#sin(π/2-x)

[zoropookie]

hi, I'm attracted to my crush Heizou like the earth is attracted to the sun— with a large force inversely proportional to the distance squared. He's the sin(90) 4 me. If we we're to be convertices, i wanna be the second convertex. to assure you that at the end of the line, i'll be there waiting for him. these formulas describe how much x²-1, if the value of x is 12. sin (π/2 – A) = cos A & cos (π/2 – A) = sin A

sin (π/2 + A) = cos A & cos (π/2 + A) = – sin Asin (3π/2 – A) = – cos A & cos (3π/2 – A) = – sin A sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A sin (π – A) = sin A & cos (π – A) = – cos A sin (π + A) = – sin A & cos (π + A) = – cos A sin (2π – A) = – sin A & cos (2π – A) = cos A sin (2π + A) = sin A & cos (2π + A) = cos A. before seeing him, my heart is a nul set, but when a vector of magnitude from his eyes at a deviation of theta radians made a tangent to my heart, it differentiated. whenever I see him, it was like, I was electrocuted with an amount of 200-1700 volts of electricity. my love for him is like the derivative of e^x, it remains constant and never change. my baby, 1+tan2A = sec2A sec2A – 1 = tan2A (1/cos2A) -1 = tan2A. putting the value of cos A = ⅘.

(5/4)2 – 1 = tan2 A tan2A = 9/16 tan A = 3/4 (sin30° + cos30°) – (sin 60° + cos60°)

= ½ + √3/2 – √3/2 – ½ = 0 If cos A = 4/5, then tan A = Cos A = ⅘. my love for him is a quadratic equation with real roots, which only he can solve by making good binary relation with me. the cosine of my love for you extends to infinity. Baby, simplify 4x³+3x²+3x-1 where x=3. and he'll know what my heart says. He is an essential to me as an element to a set. the geometry of my life revolves around his acute personality. sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)] cos(2x) = 2cos2(x)−1 = 1–2sin2(x)

tan(2x) = [2tan(x)]/ [1−tan2(x)] sec (2x) = sec2 x/(2-sec2 x) csc (2x) = (sec x. csc x)/2. my baby, when he solves this equation for "i" .

hiiii pookie how are you? i hope you're doing well!!!! kiss kiss/p

i'm not sure if this is a copypasta or not, but if it's not, that's extremely impressive and you're super smart hahahahahaha!!!

i'm doing as good as i can lol hopefully you slept well, i'm answering this after ur regular hours

[mera-mann-kehne-laga]

YAAR YE TRIGNO NHI HO RHI 😭😭😭😭😭😭😭😭😭😭😭😭😭

cos (π+x) = -cos x KAAAIIIIIIIIIISSSSEEEEEEEEEEEEEE

Aur sin(π-x) = sin x KAAAAAIIISSSEEE😭😭😭😭😭😭😭😭😭😭😭😭😭😭😭

Cos(π/2 + x) = (-sin x)?!?!?!??!?!!!!????????!(?;+-₹#+₹-@+#+@(+₹)@+219372863#+-₹#++#)@+₹)@₹+)#

[bad-dagon]

riemann spheres as a fundamental type, pt.1 basics

ive been thinking on and off about riemann spheres for a while now, a couple weeks really, and so far i think there's some utility to them as a building block of a type system of some kind for a joke/toy computer language

background: riemann spheres are a neat tool in complex analysis where we imagine a sphere whose equator intersects the complex plane, and every number on the complex plane is representable by a point on the sphere such that a line is projected from the north pole and through that point onto the complex plane. naturally, this means that the north pole is P(∞) and the south pole is P(0). see below how that would look like with other unit points of the complex numbers

a neat thing the riemann sphere allows us is to define meaningful division by zero so now z/0 = ∞ clean and simple! and also its inverse, z/∞ = 0 is well behaved as well. this simplifies doing complex analysis but stereographic projection is an absolute bitch to work with turns out, and doing arithmetic on points on the sphere is a mess because it's not a linear mapping (it's continuous though so that's fine)

if we're dealing with ONLY real numbers in relation to a circular slice of the riemann sphere, it already starts looking like a mess; for any number n∈R its projective cognate on the circle is located at (2n/n²+1, n²-1/n²+1). on the real riemann sphere though? zoo wee mamma

an arbitrary z∈C, represented as a point (x,y) on the complex plane, has to walk through a rather ugly mapping (related to the previous one) to find its point on the sphere; more accurately, given the coordinates (x,y) of the point on the plane, the point on the sphere is located at (2x/1+x²+y², 2y/1+x²+y², -1+x²+y²/1+x²+y²), which is godawful in spherical-to-polar coordinate terms, this is much simpler; for a polar pair (r,θ) the point on the unit sphere representing it is (φ,ξ) = (2*arctg 1/r, θ); and conversely projecting from the riemann sphere is also dead simple, given the zenith-azimuth pair (φ,ξ), (r,θ) = (ctg φ/2, ξ). of course, translating from polar to real coordinates is ALSO dead simple; x = r * cos θ, y = r * sin θ. if a computer system were to store complex numbers (or any coordinate on a 2d plane really), it makes sense to store them in terms of spherical coordinates of a riemann sphere, since this makes infinity well-behaved as a unit (zenith = 0, azimuth = literally who gives a fuck) and is surprisingly efficient. knowing that the zenith is ∈[0,π] and azimuth ∈[0, 2π] can allow for some formidably dumb optimisations that can save on space and ensure granularity. compared to storing them as 2d cartesian or polar coordinates, this provides the benefit of having neither number be larger than like 6.3, so an underlying/backing type that can offer great precision in this small range would be more efficient numerically than, say, floating points which have insane baggage and gaps

or iunno i'll look into that one a bit later, i'm just kind of furious right now that i rederived stereographic projection on my own when the formulas were right fucking there if id only just googled for them

[jakevor]

Hey mathblr. Today I don't have Wrong Math, but I do have a question. I was trying to find the real roots of (-1)^x. And I ended up with the formula:

n/(4k+1) where nεZ and kεZ.

I know this can make all natural powers for k=0 and it can make no powers of the type 1/(2k). But I have trouble proving, if it can be proven, that this formula can produce all the powers of 1/(2k+1) so that I can have (-1)^1/3, (-1)^1/7 and so on.

If anyone has an idea of how to prove this please help.

I will leave how I found that formula after the break.

(-1)^x=

i^2x=

exp(2(2kπ+π/2xi))=

cos(4kπ+πx)+i*sin(4kπ+πx)

For (-1)^x to have real roots the sin must be zero.

sin(4kπ+πx)=0

4kπ+πx=nπ =>

(4k+1)x=n =>

x=n/(4k+1).

[jadagul]

So i^2 is -1 (by definition), and i^3 is -i (-1 * i), and i^4 is 1 (if i*i is -1 then i*-1 is 1), and i^5 is i (1 * i), and it continues to loop from there. But how do things work for non-integer exponents? It's easy to say, of i^1.5, that it multiplied by i^0.5 is -1; but that doesn't tell us what it itself is. Do we just call it 'i^1.5' and leave it at that? Or is there some trick I'm missing that'll let us pin it down as a more traditional real or complex number?

Yeah, there's a trick; it's just dumb.

We didn't go into this in the post (I even mentioned the not-going-into-it!) but complex exponentiation works really well with e as the base, because e^(ix) = cos(x) + i sin(x). And in particular that means e^(i π/2) = i.

So if we want to compute i^(1.5) then that's e^(i π/2 * 1.5) = e^(i 3 π /4) = cos(3π/4) + i sin(3π/4) = -sqrt(2)/2 + i sqrt(2)/2.

In general if you exponentiate a complex number, you regular-exponentiate the modulus, and you multiply the angle from the positive x-axis by the exponent as well. So in your case, i has modulus 1 and angle 90 degrees, so we get the output that has modulus 1 and angle 135 degrees.

[transgenderer]

Richardson's theorem can be stated as follows:[2] Let E be a set of expressions that represent R->R functions. Suppose that E includes these expressions:

x (representing the identity function)

e^x (representing the exponential functions)

sin x (representing the sin function)

all rational numbers, ln 2, and π (representing constant functions that ignore their input and produce the given number as output)

Suppose E is also closed under a few standard operations. Specifically, suppose that if A and B are in E, then all of the following are also in E:

A + B (representing the pointwise addition of the functions that A and B represent)

A − B (representing pointwise subtraction)

AB (representing pointwise multiplication)

A∘B (representing the composition of the functions represented by A and B)

Then the following decision problems are unsolvable:

Deciding whether an expression A in E represents a function that is nonnegative everywhere

If E includes also the expression |x| (representing the absolute value function)

Deciding whether an expression A in E represents a function that is zero everywhere

If E includes an expression B representing a function whose antiderivative has no representative in E, deciding whether an expression A in E represents a function whose antiderivative can be represented in E.

------

!!! whether a combination of sins and exponents is zero is undecidable!

[itsgerges]

Best Greetings

 The Sun Is a Phenomenon, Created By The Planets Motions Energies

https://www.academia.edu/s/9c2047af81

or

https://www.academia.edu/87646952/The_Sun_Is_a_Phenomenon_Created_By_The_Planets_Motions_Energies_revised_

or

https://app.box.com/s/rdpno22picdel589y14tr2cs6wfc5957

or

https://app.box.com/s/aowanf3eyk8d4n2lnx06anyktjp62r8c

  Abstract

Paper hypothesis

The solar planets distribution between the sun and Pluto depends on one design – this design uses the distance 90000 million km as a distance unit.

The hypothesis explanation

The hypothesis tells – the planets distribution uses one design and this one design gives a chance for the planets to make their distances total equal 90000 million km for a unit period of time – there's a necessity to use this distance we explain here.

The hypothesis proof

1- Data (100733 million km = 32200 million km x π)   100733 million km        = The Planets Orbital Circumferences Total 32200 million km                    = (37100 million km -4900 million km)

37100 million km          = Pluto Orbital Circumference

4900 million km            = Jupiter Orbital Circumference

The data tells the values (100733, 37100 and 4900) depend on each other and create one design- means Jupiter and Pluto are the 2 basic points in the solar system design  

 2- Data (100733/90000) = (37100/32200)

There's an error (3%) for the data no. (2) because Pluto real distance should be Pluto Earth Distance and not Pluto orbital distance Because the solar system velocities map depend on Earth velocity – (Point no.A-3)

3-Data (100733/90000) = (120/108.2) =(170/149.6) =(629/551) =(4900/4387) = (5446/4900) = (3030/2723) = (721/655) = (4190/3717)= (5756/5127) =(5127/4495) = =(1623/1433) (Max error 2%)

100733 million km = The Planets Orbital Circumferences Total 120 million km      = Venus Mars Distance 108.2 million km            = Venus Orbital Distance 170 million km      = Mercury Mars Orbital Distance 149.6 million km            = Earth Orbital Distance 629 million km      = Earth Jupiter Distance 551 million km      = Mars Jupiter Distance 4900 million km    = Jupiter Orbital Circumference 4387 million km    = Neptune Venus Distance 5446 million km   = 2 x Earth Uranus Distance (2723 million km) 3030 million km   = Uranus Pluto Distance 721 million km     = Jupiter Mercury Distance 655 million km     = Jupiter Saturn Distance 3717 million km   = Jupiter Neptune Distance 4190 million km   = 2 x Jupiter Uranus Distance (2094 million km) (error 2.5%) 5756 million km   = Pluto Earth Distance 5127 million km   = Pluto Jupiter Distance 4495 million km   = Neptune Orbital Distance 1623 million km   = Neptune Uranus Distance 1433 million km   = Saturn Orbital Distance

4- Data (90000/4900) = (197393/10747)

197393 days        = The Planets Orbital Circumferences Total 10747 days                    = Saturn Orbital Distance   Saturn motion is important because Saturn orbital circumference =9000 million km – for that Saturn orbital period be used here.

5- Data (90000=25920 x 3.47)

25920 million km          = The distance be passed by light (300000 km/s) in one solar day

But 100733 million km x (sin 3.47 degrees) =5906 million km

5906 million km            = Pluto Orbital Distance (error 3%)

  The previous data shows 2 facts (1

st

Fact) One Map (One Design) Controls The Planets Distribution Between The Sun And Pluto – (2

nd

Fact) This One Design Uses The Distance 90000 Million Km As A Unit

 Paper first question

Why does the solar system design use the distance 90000 million km as a unit?

Let's answer in following…

If (c =300000 km/sec = light velocity) how can we measure the value (c2)?

Let's suppose we use the value (c2) for one second – how can we measure it?

In this case the value (c2) will be = 90000 million km

Now

The hypothesis tells, the planets motions aim to pass this distance (90000 million km) and to use this distance based on A Motion For A Period Of Time = One Second

(We Discover In The Paper Discussion How This Motion Can Be Created)

As a result –

The distance 90000 million km will be a equivalent to the value (c2)

And

The value (c2) is the source of energy from which the sun rays be created.

Based on that

The Sun Rays Be Created By The Planets Motions Energies Total

Means,

The Sun Is Not Doing Nuclear Fusion - Instead – the planets motions energies total be accumulated and be used as energy source to produce the sun rays- then the sun disc and body be created from the sun rays energy.  

And

Because the planets motions aim to create the sun rays they aim to pass the required distance 90000 million km and the solar system design causes this distance to be used by a motion depends on one second

 Paper second question

Can Light Beam Be Produced By Matters Relative Motion?

 The planets motions distances total per solar day (=17.75 million km)

(This distance contains the Earth moon velocity which is equal Earth velocity)

 During 1461 days the planets motions distances total be =25920 million km

Light (300000 km/s) travels in one day (86400 s) a distance =25920 million km

But the Earth Cycle is (365 +365+365+366 =1461 days)

 The idea tells us – if 1461 seconds of the Earth motion = 1 second of the sun motion – the velocities difference between the 2 motions will be= Light Velocity (300000km/s)

 The paper discusses how that can be possible- but – here – we discuss the conceptual question- can light beam be created by matters relative motion?

The planets data tells about a new method which is (Motions Use Different Rate Of Time)

The paper discussion proves the general rate of time for the planets motions which is (One Hour Of Mercury Motion Be = 24 Hours Of Any Planet Motion)

The rate of time meaning is known – and the paper discusses how this rate of time be created – but – we need to analyze its geometrical effect on planet motion –

Let's suppose it's a fact that (One Hour Of Mercury Motion Be = 24 Hours Of Any Planet Motion) – What geometrical result can be performed by that?

It's An Accumulation Of Energy – because

The planets move during (24 days) and Mercury can use the produced energy on one day- by that the rate of time causes the motion energy accumulation – also the energy accumulation be on Mercury point –means- the energy moves from the planets toward Mercury based on this rate of time

Let's ask (Why Is This Idea Strange For Us?)

Because – it suppose the energy be transported from one planet to another – it makes the solar system as a continuum of data because the data be transported among the planets – here we reach to the paper argument

 Paper Argument

The paper considers the solar system as one continuum of data – as one creature body – or one machine – the planets data be created in proportionality with one another and one law controls all planets data and one motion be planned for all planets– it's shortly – one general design controls the planets creation and motion data

The classical vision about the separated planets from one another be disproved frequently and showed it's useless because it contradicts the planets data.

The argument can be seen clearly when we ask (How Be The Matter Created?) this question disproves logically Newton theory of the sun gravity – because the planet moves by the force which caused its creation – No planet moves by the sun gravity- But

The planets motions harmony and their deep consistency of data shows that the solar system design can't be created by rigid separated bodies but by one continuum of data

 As you see 5 sticks on Earth and consider they are separated sticks from one another – in fact– the 5 sticks are 5 branches in the same one tree and they are connected by the tree root but the connection point is not seen –here – the mistake be in the description.

 The planets matters and their distances be created from One Energy – this is the main idea behind the argument – the physics accepts the matter and space be created of energies –I add one idea tells – from One Energy – the planets matters and distances be created.    

This idea connects the planets and their distances in one continuum because the energy is one – by that the planets be similar to 10 trees on a canal of water – the same one water (energy) causes their lives – the energy also be transported among the planets..

This vision gives power for the discussion ideas – because -  the rate of time will be useful here because it controls the amount of energy be sent from a point to another and that causes (redistribution) for energy which can cause (a massive energy on a point) and from this massive energy the light can be created.

Shortly

The paper provides a different vision about the solar system– because – the one energy causes connection between all planets and distances by that Pluto motion affects on Mercury motion because the solar system is one creature body

 A New Reading For The Planets Data

The continuum feature is proved by the planets data –by that- the paper vision is found basically to explain the planets data – let's summarize the proof in following

 Planet Diameter Equation (My fourth equation) proves each planet diameter be a function in its rotation period - let's summarize its concept in following

  (1) Planet Data Be Created Based On Exact Equations – As a plane or rocket manufacture – the manufacturer needs exact equations to define this plane length, width, weight and all specifications, otherwise this plane can't fly safely.   The moving planet under the physical laws has to define its diameter, mass, orbital distance, period, inclination, rotation period, axial tilt …and all data based on exact equations otherwise this planet can't move safely. I have discovered 5 equations can conclude Planets Data theoretically (2) Planet orbital distance be defined based on its neighbor orbital distance – even before this planet creation - because – the planets motions leave an empty place for the new planet –by that- each planet orbital distance be defined by its neighbor orbital distance (my first equation proves this fact)   (3) The new planet has to revolve around the sun based on its orbital distance which be defined obligatorily where no data of this planet be taken into consideration in its orbital distance definition –neither mass nor diameter –instead – the distance be defined based on its neighbor orbital distance.   But Planet diameter should be a function in its orbital distance – otherwise – this planet will be broken through its motion – The function between planet diameter and its orbital distance is a necessity to cause this planet safe motion – almost – planet mass can't cause this planet to be broken but it may decrease its velocity or create orbital inclination –the planet geometrical motion form depends on its diameter – the wrong diameter can cause this planet to be broken and destroyed. One more difficulty be found for the designer If the function contains only 2 variables which are planet diameter and its orbital  distance – in case this planet changes its orbital distance for any reason- this planet will be broken also – As a result The designer had to create a function between planet diameter and its orbital distance but also to make this function contains more variables – by that- if this planet changes its orbital distance for any reason–the other variables will be changed but the diameter will be saved – as a result- The planet diameter be created as a function in its rotation period, and the rotation period be a function in its velocity and the velocity be a function in its orbital distance   By that – the function between planet diameter and its orbital distance be created but also contains 3 more variables And If the planet changes its orbital distance – its rotation period, orbital period and velocity will be changed but the diameter will be saved. Planet diameter equation (my fourth equation) proves planet diameter be a        function in its rotation period and the planet rotation period be a function in its velocity Kepler laws prove (Planet Velocity Be A Function In Its Orbital Distance) We discuss Planet diameter equation in Point No. (E) But – Why Does The Equation Work Sufficiently Through The Planets Data? How can that be possible "each planet diameter be a function in its rotation period?" This result needs a continuum of data to be performed because its one force passes through all planets and forces their data to be created as a function in their rotation period- The meaning is seen clearly If the data can't be transported among the planets the equation can't work The equation works because the solar system is on A Continuum Of Data The planets data be created as one group of data controlled by one law and can be transported among the planets and affects on each other.

 Notice

Planet orbital distance equation (my first equation) doesn't need the continuum of data because each planet orbital distance depends on its neighbor – here we deal with one building be consisted of stories – each story affects on its neighbor – but planet diameter equation necessitates to have one force passes through all planets data – this is the basic effect of the continuum data – it's similar to groundwater under 10 houses all of them suffer from the same force and effect -      

The machine real motion for the distance 90000 million km

The machine motion depends on a period 2 seconds, for that, the planets move a total distance =180000 million km

Saturn moves (90000 million km) depends on Uranus motion support and

Jupiter moves (90000 million km) depends on All planets motions support

As a result

The outer planets define their orbital distances based on Saturn orbital distance

And

The inner planets define their orbital distances based on Jupiter orbital distance

That causes the solar planets distribution depends on one design and this design uses the distance 90000 million km as a distance unit.

 The Solar System Creation Theory (A Summary)

The planets matters and their distances be created of one energy – this one energy be provided by one light beam– this light beam travels with a velocity=1.16 million km/s  

Matter creation process is done as following

The process input is (one light beam its velocity 1.16 million km/s)

The process output is (matter + space + one light beam its velocity 300000 km/s)

The energy of the original light beam 1.16 million km/s be consumed in the creation and the light known velocity 300000 km/s be produced as a side product –

  Let's analyze the creation forma in following (1.16/0.3) x 2π =24.3 (a) What's this value (24)?? It's the rate of time (One Hour Of Mercury Motion =24 Hours Of Any Planet Motion) Why is this rate so necessary? Because One Second Of light (300000 km/s) Be = One Solar Day Of Any Planet Motion – And One Second Of light (1160000 km/s) Be = One Solar Day Of Any Planet Motion –

Means

The produced matter moves with its parent light beam by using a different rate of time which is (One Second Of Light Motion = One Day Of Planet Motion) That causes the necessity of the time rate (1=24) That shows the force which created the planet causes its motion (b) What's this rate (2π)? This rate (2π) causes all motions in the universe to be in circular and elliptical forms – it's a result of the matter creation out of light beam (c) The matter creation be done by the light beam its velocity 1.16 million km/sec – but this light beam energy be consumed in the creation process – that means- any new produced matter needs a new light beam its velocity 1.16 million km/sec – means- the planets motions energies accumulation produces the sun rays (300000 km/s) and also produces the original light beam (1.16 million km/sec) without which no matter can be created.

Paper objective

The paper proves the fact (The Sun Rays Be Created By The Planets Motions) and from the sun rays energy the sun disc and body be created

Means

The sun is a phenomenon created by the planets motions energies and the sun life depends on a cycle because its existence depends on the planets motions which be done in cycles.                                      

Gerges Francis Tawdrous +201022532292

Physics Department- Physics & Mathematics Faculty  

Peoples' Friendship university of Russia – Moscow   (2010-2013)

 Curriculum Vitae           https://www.academia.edu/s/b88b0ecb7c

E-mail                            mrwaheid@gmail.com, mrwaheid1@yahoo.com

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[tutoroot]

What are the List of Trigonometric Formulas? 

Trigonometry is the study of triangles and connections between triangle lengths and angles in mathematics. Trigonometric formulas and a list of trigonometric identities form one of the most timeless and important facets of mathematics.

Trigonometry and related equations have a plethora of applications. Triangulation, for example, is used in Geography to calculate the distance between landmarks; in Astronomy they are used to determine the distance to neighboring stars; and, in satellite navigation systems. In many other ways, Trigonometric formulae are useful and indispensable too.

Trigonometry Formulas

Trigonometry formulas are a collection that uses trigonometric identities to solve problems, involving the sides and angles of a right-angled triangle. For given angles, these trigonometry formula include trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent. While the trigonometric formulae involving trigonometric identities are the core of the subject, we also would like to understand the importance of trigonometric identities, which in a basic sense refers to an equation that involves trigonometric ratios of an angle.

In the following sections, trigonometric identities, including Pythagorean identities, product identities, co-function identities (shifting angles), sum & difference identities, double-angle identities, half-angle identities, and so on are explained in detail.

List of Trigonometric Formulas

When we first learn about trigonometric formulas, we only consider right-angled triangles. As we know, a right-angled triangle has three sides: the hypotenuse, the opposite side (perpendicular), and the adjacent side (Base). The longest side in a right-angled triangle is known as the hypotenuse, the opposite side is perpendicular, and the adjacent side is where both the hypotenuse and the opposite side rest. These sides and the basic structure of the right-angled triangle go a long way in determining the depth of understanding of trigonometry formulae. In short, the right-angled triangle is the reference point to derive or arrive at trigonometry formulae or trigonometric identities.

Basic Trigonometric Formulas

In Trigonometry, there are six ratios that are utilized to find the elements. They are referred to as trigonometric functions. Sine, cosine, secant, cosecant, tangent, and cotangent are the six trigonometric functions.

Inverse Trigonometric Formulas

Trigonometric ratios are inverted using inverse trigonometry formulas to produce inverse trigonometric functions such as sin θ = x and θ=sin−1x. In this case, x can take the form of whole integers, decimals, fractions, or exponents. 

Trigonometry Identities

Trigonometric Identities are equalities that involve trigonometry functions that stay valuable for all variables in the equation.

There are several trigonometric identities relating to the side length and angle of a triangle. These identities stay true to the right-angle triangle.

As explained, these are all derived from a right-angled triangle. If we know the height and base side of the right triangle, it will become easier to know sine, cosine, tangent, secant, cosecant, and cotangent values, by applying trigonometric formulas. We can also derive reciprocal trigonometric identities by applying trigonometric functions.

Periodicity Identities

The periodicity identities are formulas used to shift the angles by π/2, π, 2π, etc. They are also classified under cofunction identities.

If one observes keenly, fundamentally, all trigonometric identities are cyclic. They repeat after this periodicity constant. The periodicity constant varies among the trigonometric identities and is different for each.

Trigonometric Identities of Opposite Angles

As we dwell deep into trigonometry formulas and various other aspects of this branch of mathematics, we explore more interesting features that enhance our subject knowledge and take us through new paths of knowledge. One such is the trigonometric identities of opposite angles, where, a trigonometry angle that is measured in its clockwise direction, is measured in negative parity. The trigonometric ratios for the angle’s negative parity are as follows:

Complementary Angles Identities

As the expression suggests, complementary angles are the pair of angles whose added measure comes to 90°.

Supplementary Angles Identities

These are a pair of angles whose measure adds to 180°.

At  Tutoroot, we offer personalised trigonometry tutoring to ensure a clear understanding. Our expert instructors use simple teaching approaches for an effective understanding of the subject. Sign up with Tutoroot’s Online Tuition for Maths to learn more.

[dandream235]

Ficha de información

Nombre: Danilo, o simplemente Dan

Edad: 20 años

Cumpleaños: 27 de Agosto, nacido en 2004

Sexo: Hombre

País: Colombia (les invito una empanada, y natillas para navidad)

Estado actual: Soltero, estudiante universitario, Ing de sistemas

Proyectos actuales: [W.I.P.]

[Stand by indefinido]

["Project Rocky Hds" ; "Project VTC" ; "Project HC" ; "Project AlterTrack" ]

[Stand by parcial]

["Project Deep Shadows" ; "Project Soaring Roar"]

+ Cuenta Pricipal de desarrollo de proyectos y contenido +

@dandream235

+ Cuenta Secundaria personal de expresión de ideas y opiniones +

@danthinker

Redes sociales:

- Twitter (X) -

+ Cuenta principal : @ DanDream235

+ Cuenta secundaria / política : @ Danilojcm_col

+ Cuenta de DanDev (T) : @ FNFGFSingsBack

[ Cuenta principal para FNF, donde su iré el contenido de 'Girlfriend Sings Back' (tengo pendiente averiguar que otro nombre ponerle, pues... va a haber un remake, pero a lo grande!) y 'AlterTrack' ]

(Tengo planeado hacerlo, pero no sé todavía como, y tampoco si lo haría solo o con más personas)

- Discord -

Única Cuenta : dandream235 / DanDiscDream#9262

- Wattpad -

Única cuenta : @ DanDream235

- YouTube -

Única cuenta : @ dandream235

(Más que todo la cuenta, todavía no se que hacer con el canal)

- Ko-Fi -

+ DanDream235 +

+ Cuenta "Principal" : @dandream235

(Todavía no la uso)

Usernames y usuarios de videojuegos:

- Minecraft -

+ DanDream235 +

( A veces varía, uso launchers No Premium )

[30/08/2023] Compré Minecraft Premium :D

[08/03/2024] Me añaden? podemos jugar :)

- Steam -

+ DanDream235 +

(Juegos con Steam : Terraria, TModLoader, UNDERTALE, DELTARUNE, PUBG, ARK, BattleBlock Teather, Call of Duty, CS:GO, Geometry Dash, Poly Bridge 1 y 2, Dungeon Defenders, Cobalt, Castle Crashers, Human Fall Flat, Plague Inc, Left 4 Dead 2)

- Geometry Dash -

+ SealedVenus +

(Larga historia)

- Squad Busters -

+ DanDream235 +

- Roblox -

+ SealedVenus235 / DanBloxDream +

(Si, larga historia, algún día la contaré)

- Xbox -

+ DanDream235 +

(Pa' cuando repare mi Xbox 360 o me consiga una One o una Series S, π π π π)

Trataré de actualizarlo con el paso del tiempo, e intentaré no borrar links viejos, solo por si acaso

Redes Sociales "eliminadas":

Una cuenta de Twitter llamada @ FNFDisbeliefGF , un proyecto cancelado que en serio me arrepiento de la idea inicial con la que lo quería hacer. Tal vez a futuro haga una reimaginación del concepto, solo que menos cringe :p . Me olvidé de la contraseña y no se si pueda volver a acceder. Link :

(Edit 13/09/2023): Si tengo el acceso, solo que lo había olvidado xd, igualmente no creo hacer algo con esa cuenta

(Edit 08/03/2024): No tengo el acceso, y Twitter no colabora, tmr

(Edit 27/04/2024): Sigo intentando, pero carajo, todo por el operador del número, me cago en la la put-

La cuenta de FNF' AlterTrack: @ FNFAlterTrack ha sufrido una especie de Shadowban o algo así, ya que tiene alcance limitado, por lo que quedará algo archivada, a menos que claro, se reviva con el desarrollo del mod

La anterior cuenta de @ DanDevDream de Twitter/X ha sido, por decirlo así, desabilitada, en el sentido de ya no ocuparla más, pues por temas personales que creo puse en mis últimos tweets, decidí empezar de 0 en una nueva, la cual está en mi biografía en DanDevDream

(Edit 27/04/2024): Sufrió un shadowban, carajo

[shuxiii]

f(x) = x^2/3 + e/3(π-x^2)^1/2 sin(απx)

i-i-i-i think u failed to shape a heart

[erikabsworld]

Mastering Numerical Differentiation: A Step-by-Step Guide with a Tough University-Level Assignment

Welcome to the fascinating world of numerical differentiation, where mathematical concepts meet real-world applications. In this blog, we will delve into a challenging university-level assignment on numerical differentiation. If you've ever felt overwhelmed by the complexities of this topic, fear not! We'll break it down, offering a comprehensive explanation and step-by-step guide to tackle a tough assignment question.

Understanding Numerical Differentiation:

Numerical differentiation is a mathematical technique used to estimate the derivative of a function at a particular point. Unlike analytical methods that involve intricate formulas, numerical differentiation relies on approximation methods to compute derivatives. One popular approach is the finite difference method, which involves finding the slope of a tangent line using small intervals.

Sample Assignment Question:

Consider the function f(x) = sin(x) + x^2. Find the derivative of f(x) at x = π using numerical differentiation with a step size of h = 0.1.

Step-by-Step Guide:

Step 1: Define the Function Begin by identifying the function you're working with. In our example, f(x) = sin(x) + x^2.

Step 2: Choose a Step Size (h) Select a small step size (h) to approximate the derivative. In this case, h = 0.1.

Step 3: Use the Finite Difference Method Apply the finite difference method to find the slope at x = π. Use the formula:

f′(x)≈ [f(x+h)−f(x)]/h .

Step 4: Substitute Values Plug in the values into the formula:

f′ (π)≈ [f(π+0.1)−f(π)]/0.1.

Step 5: Calculate the Result Compute the numerical derivative using the values from Step 4.

Website Plug: How We Help Students with Numerical Differentiation Assignments

Struggling with numerical differentiation assignments? Our website, matlabassignmentexperts.com, is here to provide the best numerical differentiation assignment writing help. Our team of skilled mathematicians and writers are well-versed in numerical differentiation and can guide you through challenging assignments. We offer personalized solutions, ensuring that you not only understand the concepts but also excel in your coursework.

Conclusion:

Numerical differentiation may seem daunting at first, but with the right approach, it becomes a manageable and valuable skill. By following the step-by-step guide provided in this blog, you can confidently tackle university-level assignments and gain a deeper understanding of numerical differentiation. Remember, if you ever need additional support, matlabassignmentexperts.com is just a click away, ready to assist you on your academic journey.

[capybaraonabicycle]

ohh i have a question for you, bc i am dumb and had to review shit i'd done a While ago so when it comes to trig functions, what the heck does sine cosine and tangent MEAN? like, i remember being taught how to do it, just not why and i feel like that was smth i needed to learn cause im like. 90% sure i failed trig lol

Okay, I love this, this is MY question. Because I was very lost on sine & Co in school, but now they feel like brothers to me. I had a very good teaching assistant telling us 4 ways to define them and then he said to pick the one that felt most natural to us. And that really helped.

None of the ways are super simple, but I think they are all very useful under different circumstances and reviewing all together helps getting a feeling for the functions. The fourth definition will probably be most familiar to you and hence the easiest, so wait for it if you get lost before <3 This is also why I will keep the other three definitions rather short. Please ask if you'd like to know more!

And like, on the why, before we start : some definitions have different obvious nifty uses (will always highlight them in blue in the end) but generally, we just define these functions because we can? Like that's what math is to me. You look at some object/concept/whatever and go 'my, isn't that pretty, let's see what else it can do'. So here we will define four times some functions with properties we want because that's just your good old mathematical fun :) (this is not scientific talk, this is how I understand maths - I feel like I should put that as disclaimer on every 2nd sentence tbh 😅)

Also, easiest things first:

Tangent is just sine divided by cosine. That's literally the definition, that's all it is. Tangent's definition isn't magic, tangent's definition is super simple and loves you :)

tan(x) := sin(x) /cos(x)

(also a disclaimer: I am being VERY lenient with definitions of functions here. It is not mathematically exact everywhere, just like you might find it in a class in school tbh, but I decided I didn't want to confuse you further and again, not so sure how familiar with functions you are)

So, without further ado, let's see how many definitions of the trigonometric functions I remember correctly:

(I am pretty sure Nr 1 is the one my teacher started with but you might want to completely skip that one - you'll see why.)

Cosine is the first/easiest function that fulfills the differential equation:

y+y''=0, y(π/2)=0 and y(x) ≠0 for all x on [0,π/2)

So this means we look for a function y that solves the demands

The second derivative of y is the negative of the function y.

π/2 is the 'first' root of y. That is if we plug π/2 into y we get 0 and we don't have any smaller positive number which gives us 0.

The differential equation is solved by multiple versions of sine and cosine, but the simplest that also has π/2 as smallest root is cosine itself. Sine then is just the negative derivative of cosine.

WHY (do we want this definition/the functions that solve this) : Differential equations are extremely important especially in physics. And y+y"=0 is basically as simple as it gets with differential equations, it'd be great to have a nice name for plus understanding of the functions that solve it

2. The Taylor expansion / sum formula

The sine and cosine functions can be written via sequences (infinite sums) *(footnote 1)* as:

sin(x) := Σ_(n=0)^(infinity) [((-1)^n)/((2n+1)!) * x^(2n+1)] = x - x^3/3 + x^5/5 - x^7/7± cos(x) := Σ_(n=0)^(infinity) [((-1)^n)/((2n)!) * x^(2n)] = 1 - x^2/2 + x^4/4 - x^6/6 ±

Hang on, Wikipedia does that more prettily:

So in a way, you could interpret the trigonometric functions as 'infinite polynomials'. This name is mathematically wrong/ugly - they are NOT polynomials - but if it helps feeling closer to them: go for it.

This is actually why we can approximate the trigonometric functions by polynomials this well, something that is used a lot in numerics (I think) and engineering (I know). You might have seen physicists (or similar) say cosine is 'almost 1' for small x or sine is 'almost linear' for small x. The sum formula is why.

WHY: polynomials and converging sequences rule basically, not sure whether there is a better explanation for this one

3. The Euler approach

Sine and cosine are defined by the behaviour of the Euler number e. Do you know it? It is a constant, real number approximately equal to

e≈2.71828.

It has the property that the function e^x is its own derivative, that is that the slope of e^x is always exactly the same as the value 🤯 I know, awesome.

If we now consider the function

e^(ix)

with i the imaginary unit and x in the real numbers, we have the identity

e^(ix)=cos(x) +i*sin(x).

So cosine and sine are in a way parts of the Euler function.

(this is actually linked to the second point bc we can get the sum expansions from the sum expansion of e. And it is linked to the first point bc it tells us the thing about sin(x)"=-sine(x) and accordingly for cosine.)

Okay, so I get how this might not seem very helpful either if you aren't familiar with the Euler number or complex numbers. But as above, most other definitions follow from this one so once you get some understanding of both, this is actually THE definition that gives you the most security :)

WHY: to get the other definitions easily from this. Also, as you can see if you know a little bit about complex numbers, the trigonometric functions give us the real and imaginary part of the Euler function. Which is awesome to have as humans prefer to thing in real numbers to complex ones

4. Angles aka the definition they should teach in a good school

So, this one is easiest to understand, I think, at least if you're a visual learner. So definitely the hardest for me to explain with words 😅

The sine function and cosine function are defined as giving the (oriented) lengths of the sides of a right-angle triangle whose hypotenuse is the radius of the unit circle.

Like this:

So you might remember from school - and I've kinda been using it above in secret before - that sine and cosine take an angle as their argument. That is the angle between the x-axis and the hypotenuse (black line that ends at orange dot) in the above picture. It is marked by a dotted arc. As the angle grows bigger (takes any value between 1 and 2π *(footnote 2)* ) the triangle will change.

The sine function will always give us the (oriented) length of the dotted line down from the orange dot to the x-axis; that is the y-value of the orange dot.

The cosine function will always give us the (oriented) length of the remaining side of the triangle (on the x-axis); that is the x-value of the orange dot.

How do we use this in school? You might have heard the

sin(α) = opposite/hypothenuse cos(α) = adjacent/hypothenuse

story. Well, this is simple to understand: What I have (kinda) told you is

sin(α) = opposite cos(α) = adjacent

(check this :) ), but in this definition, the length of the hypotenuse is 1! So it simply cancels in the above equation. And if we have a different size triangle with hypotenuse ≠1, we only need to scale by that factor to get to the definition, hence the formula from school!

WHY: The formula from school, basically. Easy way to calculate angles and side lengths of a right-angled triangle. Very useful in basic engineering

Footnote 1 on sums:

If you're unfamiliar with sums and the Σ (sigma), ask me again, I can explain. (Also, I actually did explain this in the post I wrote for Pine's art project already. I think I sent you that one?) Or read here (at Notation/Capital Sigma Notation), I think it is simple-ish to understand. Then again, I have been using the Σ on the daily for years.

Footnote 2 on angles and numbers:

In their 'natural' state, the trigonometric functions are defined on the real numbers, so any number in the interval (-infinity, infinity). They are 2π-periodic, which corresponds to going once around a circle.

2π is just some real number, in fact it is the circumference of the unit circle (radius=1), so it makes sense to understand a circle as 'being 2π', and any angle some fraction of the circumference (exactly the length we cut off with the hypotenuse to get that angle).

In school, we often learn of a circle being 360°. I honestly don't know why and by now that way of thinking is alien to me. But it might feel more natural to you.

Sine and cosine - to my knowledge - are not defined for degrees! They are defined for real numbers. So to use them with degrees on a circle, you always have to calculate back and forth.

A good calculator can do that for you - you should be able to switch sine and cosine between degrees (DEG) and radiant(RAD) in the settings. But it isn't too difficult to do by hand either. Just start with

360° '=' 2π or 180° '=' π

and go from there.

In a nutshell: sine and cosine are super useful for maths and science for several reasons and it's not just about triangles! They have at least four equivalent definitions and any one can be used at any given moment, whatever suits your needs best :) If you want to go into higher maths or engineering or physics, they will be your best friends

Funfact that you probably know already: I mentioned that these functions are periodic, yet that isn't even (outrightly) asked for in the definitions. Still, it is one of their most useful qualities, making them the perfect functions to describe waves with - which we need for anything to do with fluids or light -> super useful for physics, engineering and medicine (just think about heartrates and brainwaves)

I hope this was helpful in some way? Feel free to ask whatever :)

[quantumstudy]

Q: Find the range of the function y = 3 sinx + 4 cos(x + π/3) + 7

Solution : $\displaystyle cos (x + \frac{\pi}{3}) = cos x cos\frac{\pi}{3} - sinx sin\frac{\pi}{3} $

$\displaystyle cos (x + \frac{\pi}{3}) = \frac{1}{2} cos x - \frac{\sqrt{3}}{2} sinx $

$\displaystyle y = 3 sinx + 4 (\frac{1}{2} cos x - \frac{\sqrt{3}}{2} sinx) + 7 $

$\displaystyle y = 3 sinx + 2 cos x - 2 \sqrt{3} sinx + 7 $

$\displaystyle y = ( 3 - 2 \sqrt{3} ) sinx + 2 cos x + 7 $

Put $\displaystyle 3 - 2\sqrt{3} = r cos \alpha$ ...(i)

and $ 2 = r sin\alpha $  ...(ii)

$\displaystyle y = r sinx . cos \alpha + r cos x . sin\alpha + 7 $

$\displaystyle y = r sin(x + \alpha) + 7 $

Range of sin(x+α) is -1 to 1

Range of y = -r + 7 , r + 7

Now by squaring & adding equation (i) & (ii) we can get the value of r .

[phuongdg]

Công thức tính diện tích xung quanh của hình nón, CT dễ hiểu

Bạn đang cần sử dụng công thức tính diện tích xung quanh của hình nón để làm bài tập liên quan nhưng chưa kịp nhớ ra? Vậy mời các bạn cùng tham khảo bài viết dưới đây của muahangdambao.com để biết công thức cũng như cách tính diện tích xung quanh của hình nón nhé!

Hình nón là hình gì?

Trước khi đi vào tìm hiểu công thức tính diện tích xung quanh của hình nón, chúng ta cần nắm rõ hình nón là hình gì.

Hình nón là một hình học không gian tương đối đặc biệt Trong Toán học, hình nón là một dạng hình học không gian ba chiều. Hình học này có một mặt phẳng gọi là đáy và một mặt cong hướng lên phía trên được gọi là đỉnh. Trong đời sống hằng ngày, các bạn học sinh có thể dễ dàng bắt gặp các ứng dụng có thiết kế hình nón như nón lá, vỏ của cây kem ốc quế, mũ sinh nhật,…

Tính chất cơ bản cần ghi nhớ của hình nón là gì?

Để tính diện tích xung quanh hình nón bạn cũng cần biết các tính chất cơ bản, đó là: Hình có một đỉnh chính là hình tam giác Đáy của hình nón luôn là hình tròn Hình nón là hình rất đặc biệt, không hề có bất cứ cạnh nào Khoảng cách tính từ tâm của vòng tròn đáy cho đến đỉnh được gọi là chiều cao (h) của hình nón. Trong hình nón, hình được tạo bởi bán kính đáy và đường cao chính là một tam giác vuông.

Các loại hình nón cơ bản thường gặp

Căn cứ vào hình dạng cũng như vị trí của đỉnh nón, hình nón sẽ bao gồm những loại như sau: Hình nón tròn: Là kiểu hình nón có phần đỉnh nằm thẳng lên trên và vuông góc với mặt đáy.  Hình nón xiên: Là kiểu hình nón có phần đỉnh nằm xiên và không vuông góc với mặt đáy. Hình nón cụt: Là dạng hình nón bị cắt mất phần đỉnh. Giống như khi hình nón bị cắt bởi một mặt phẳng song song với phần đáy, phần mặt phẳng nằm trong hình nón sẽ là một hình tròn. Hình tính từ phần đáy và mặt phẳng hình tròn này còn được gọi là hình nón cụt.

Có 3 loại hình nón cơ bản là nón tròn, nón xiên và nón cụt

Công thức tính diện tích hình nón là gì?

Diện tích hình nón sẽ bao gồm diện tích xung quanh và diện tích toàn phần của nó. Công thức và cách tính cụ thể sẽ được chúng tôi gửi đến bạn ngay sau đây: Công thức tính diện tích xung quanh hình nón là gì? Diện tích xung quanh hình nón chỉ bao gồm diện tích bề mặt bao quanh hình nón, không bao gồm phần diện tích mặt đáy. Diện tích xung quanh hình nón công thức cụ thể như sau:  Sxq (hình nón) = π x R x L.

Ghi nhớ công thức để tính diện tích xung quanh hình nón Cụ thể trong đó: Sxq: Là diện tích xung quanh hình nón (cm2) π: Là hằng số Pi, với Pi được tính xấp xỉ bằng 3,14 R: Là bán kính của phần đáy hình nón. L: Là đường sinh của hình nón. Công thức tính diện tích toàn phần của hình nón là gì? Diện tích toàn phần hình nón chính là độ lớn của toàn bộ không gian hình bị chiếm giữ, bao gồm cả diện tích xung quanh và diện tích của hai đáy tròn. Do đó, để tính được diện tích toàn phần của hình nón, chúng ta cần tính được tổng diện tích xung quanh hình nón bằng công thức trên và diện tích của đáy. Cụ thể là: Stp = Sxq +  S(đáy) = (π x R x L) + π x R^2.

Hướng dẫn chi tiết cách tính diện tích xung quanh hình nón

Dựa theo công thức tính diện tích xung quanh của hình nón, để tính diện tích xung quanh các bạn cần biết được bán kính mặt đất của hình nón và độ dài đường sinh của hình nón.

Tìm đầy đủ các dữ kiện sau đó áp dụng công thức tính diện tính Đầu tiên các bạn cần tính ra bán kính R của mặt đáy hình nón nếu chưa biết. Tiếp theo các bạn tính đến độ dài đường sinh L của hình nón nếu chưa biết. Sau khi đã biết được hai dữ kiện R và L các bạn chỉ cần áp dụng công thức Sxq = π x R x L để tính ra diện tích xung quanh hình nón. Ví dụ: Cho hình nón có góc ở đỉnh là 120 độ, độ dài của đường sinh là 20 cm. Yêu cầu tính diện tích xung quanh hình nón nói trên. *Lời giải: Gọi đỉnh của hình nón là O, tâm đáy là H. Kẻ một đường thẳng đi qua tâm đáy AB (đường kính đáy). Như vậy góc AOB = 120 độ => góc AOH = 60 độ, OA= OB = 20cm.  Xét tam giác OHA có: R = HA = OA x sin(góc AOH) = 20 x sin60 = 20 x (3/2) = 103. Tư đây, ta tính được diện tích xung quanh hình nón là: Sxq = π x R x L = π x 103 x 20 = 2003π (cm2). Có thể bạn sẽ cần: Hình lập phương là gì? Công thức tính chu vi, diện tích và thể tích Hình thoi là gì? Dấu hiệu nhận biết và công thức tính diện tích, chu vi Như vậy trên đây, trang tin muahangdambao.com đã chia sẻ đến các bạn công thức, cách tính và ví dụ cụ thể cách tính diện tích xung quanh của hình nón. Hy vọng qua bài viết này sẽ giúp các bạn học sinh giải bài tập nhanh hơn. Read the full article

[avsarhub]

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Quadrature Carrier Multiplexing

A Quadrature Carrier Multiplexing (QCM) or Quadrature Amplitude Modulation (QAM) method enables two DSBSC modulated waves, resulting from two different message signals to occupy the same transmission band width and two message signals can be separated at the receiver. The transmitter and receiver for QCM are as shown in figure

3.1.

Figure 3.1: QCM transmitter and receiver

The transmitter involves the use of two separate product modulators that are supplied with two carrier waves of the same frequency but differing in phase by -90o. The multiplexed signal s(t) consists of the sum of the two product modulator outputs given by the equation 3.1.

---------- (3.1) s(t) = Acm1(t)cos(2πf tc )+ Acm2(t)sin(2πf tc )

where m1(t) and m2(t) are two different message signals applied to the product modulators. Thus, the multiplexed signal s(t) occupies a transmission band width of 2W, centered at the carrier frequency fc where W is the band width of message signal m1(t) or m2(t), whichever is larger.

At the receiver, the multiplexed signal s(t) is applied simultaneously to two separate coherent detectors that are supplied with two local carriers of the same frequency but differing in phase by -90o. The output of the top detector is 12 Acm1(t) and that of the bottom detector is 12 Acm2(t).

For the QCM system to operate satisfactorily, it is important to maintain correct phase and frequency relationships between the local oscillators used in the transmitter and receiver parts of the system.

Hilbert transform

The Fourier transform is useful for evaluating the frequency content of an energy signal, or in a limiting case that of a power signal. It provides mathematical basis for analyzing and designing the frequency selective filters for the separation of signals on the basis of their frequency content. Another method of separating the signals is based on phase selectivity, which uses phase shifts between the appropriate signals (components) to achieve the desired separation.

In case of a sinusoidal signal, the simplest phase shift of 180o is obtained by “Ideal transformer” (polarity reversal). When the phase angles of all the components of a given signal are shifted by 90o, the resulting function of time is called the “Hilbert transform” of the signal.

Consider an LTI system with transfer function defined by equation 3.2.

− j, f > 0

H( )f =0, f = 0

----------------- (3.2) j, f < 0

1, f > 0

and the Signum function given by sgn( )f =0, f = 0

−1, f < 0

The function H(f) can be expressed using Signum function as given by 3.3.

H(f )=− jsgn(f )

--------------------- (3.3) We know that , and . 1e− jπ2 =− j 1e jπ2 = j e± jθ= cos(θ)± jsin(θ)

Therefore, 1e− jπ2, f > 0

H( )f = π



j

1e 2, f < 0

Thus the magnitude for all H(f ) =1, f, and angle

−π , f > 0

( )f = 2

∠H



+π2, f < 0

The device which possesses such a property is called Hilbert transformer. When ever a signal is applied to the Hilbert transformer, the amplitudes of all frequency components of the input signal remain unaffected. It produces a phase shift of -90o for all positive frequencies, while a phase shift of 90o for all negative frequencies of the signal.

If x(t) is an input signal, then its Hilbert transformer is denoted by xˆ(t) and shown in the following diagram.

To find impulse response h(t) of Hilbert transformer with transfer function H(f).

Consider the relation between Signum function and the unit step function.

sgn(t)= 2 u(t)−1= x(t),

Differentiating both sides with respect to t,

d

{x( )t }=2δ( )t dt

Apply Fourier transform on both sides,

sgn( )t ↔ 2 sgn( )t ↔ 1 jω jπf

Applying duality property of Fourier transform,

1

−Sgn( )f ↔ jπt

We have

H(f )=− jsgn(f )

H( )f ↔

Therefore the impulse response h(t) of an Hilbert transformer is given by the equation 3.4,

h( )t = -------------------- (3.4)

Now consider any input x(t) to the Hilbert transformer, which is an LTI system. Let the impulse response of the Hilbert transformer is obtained by convolving the input x(t) and impulse response h(t) of the system.

xˆ(t)= x(t )∗h(t) xˆ( ) ( )

xˆ( ) dτ -------------------- (3.5)

The equation 3.5 gives the Hilbert transform of x(t).

The inverse Hilbert transform x(t) is given by

x( ) dτ -------------------- (3.6)

We have xˆ(t)= x(t)∗h(t)

The Fourier transform Xˆ (f ) of xˆ(t) is given by

Xˆ (f )= X(f )H(f )

Xˆ (f )=− jsgn(f )X(f ) -------------------- (3.7) Applications of Hilbert transform

1. It is used to realize phase selectivity in the generation of special kind of modulation called Single Side Band modulation.

2. It provides mathematical basis for the representation of band pass signals.

Note: Hilbert transform applies to any signal that is Fourier transformable.

Example: Find the Hilbert transform of . x(t)= cos(2πf tc )

1

X( )f = [δ(f − fc )+δ(f + fc )]

2

Xˆ (f )=− jsgn(f )X(f )

ˆ ( )f =− jsgn( )f 1 [δ(f − fc )+δ(f + fc )]

X

2

Xˆ ( )f = 1 [δ(f − fc )−δ(f + fc )]

2 j

Xˆ ( )f = 1 [δ(f − fc )−δ(f + fc )]

2 j

This represents Fourier transform of the sine function. Therefore the Hilbert transform of cosine function is sin function given by

xˆ(t)= sin(2πf tc )

Pre-envelope

Consider a real valued signal x(t). The pre-envelope x+(t)for positive frequencies of the signal x(t) is defined as the complex valued function given by equation 3.8.

x+(t)= x(t)+ jxˆ(t) ------------------ (3.8)

Apply Fourier transform on both the sides,

X+(f )= X(f )+ j[− jsgn(f )X(f )]

2X(f ), f > 0

X+( )f =X( )0 , f = 0 -------------------- (3.9)

0, f < 0

The pre-envelope x−(t) for negative frequencies of the signal is given by

x−(t)= x(t)− jxˆ(t)

The two pre-envelopes x+(t) and x−(t) are complex conjugate of each other, that is x+(t)= x−(t)*

The spectrum of the pre-envelope x+(t) is nonzero only for positive frequencies as emphasized in equation 3.9. Hence plus sign is used as a subscript. In contrast, the spectrum of the other pre-envelope x−(t) is nonzero only for negative frequencies. That

is

0, f > 0



X−( )f =X( )0 , f = 0 ------------------- (3.10) 2X( )f , f < 0

Thus the pre-envelopes x+(t) and x−(t) constitute a complementary pair of complex valued signals.

W f

Figure 3.2: Spectrum of the low pass signal x(t)

Properties of Hilbert transform

1. “A signal x(t) and its Hilbert transform xˆ(t)have the same amplitude spectrum”.

The magnitude of –jsgn(f) is equal to 1 for all frequencies f. Therefore x(t) and xˆ(t) have the same amplitude spectrum.

That is Xˆ (f ) = X(f ) for all f

2. “If xˆ(t) is the Hilbert transform of x(t), then the Hilbert transform of xˆ(t), is –x(t)”.

To obtain its Hilbert transform of x(t), x(t) is passed through a LTI system with a transfer function equal to –jsgn(f). A double Hilbert transformation is equivalent to passing x(t) through a cascade of two such devices. The overall transfer function of such a cascade is equal to

[− jsgn( f )]2 =−1 for all f

The resulting output is –x(t). That is the Hilbert transform of xˆ(t) is equal to –x(t).

Canonical representation for band pass signal

The Fourier transform of band-pass signal contains a band of frequencies of total extent 2W. The pre-envelope of a narrow band signal x(t) is given by

x+(t)=~x(t)e j2πf tc , ---------------------- (3.11) where ~x(t) is complex envelope of the signal x(t).

Figure 3.6: spectrum of the complex envelope ~x(t)

Figure 3.4 shows the amplitude spectrum of band pass signal x(t). Figure 3.5 shows amplitude spectrum of pre envelope x+(t). Figure 3.6 shows amplitude spectrum of complex envelope ~x(t).

Equation 3.11 is the basis of definition for complex envelope ~x(t) in terms of pre-

envelope x+(t). The spectrum of x+(t) is limited to the frequency band fc − w < f < fc + w as shown in figure 3.5. Therefore, applying the frequency-shift property of Fourier transform to equation 3.11, we find that the spectrum of the complex envelope ~x(t) is limited to the band − w < f < w and centered at the origin as shown in figure 3.6. That is, the complex envelope ~x(t) of a band pass signal x(t) is a low-pass signal.

Given signal x(t) is the real part of the pre-envelope x+(t). So, we express the original band pass signal x(t) in terms of the complex envelope ~x(t), as follows

x(t)=Re [~x(t)e j2πfct ]

------------------- (3.12)

But ~x(t) is a complex quantity.

~x (t)= xI (t)+ jxQ( t) ------------------- (3.13) where and are both real valued low pass functions. This low pass property is xI (t) xQ(t)

inherited from the complex envelope ~x(t). Therefore original band pass signal x(t) is expressed in canonical form by using equations 3.12 and 3.13 as follows.

x t( )= Re [~x(t)e j2πf tc ]

x(t)=Re[xI ( t)e j2πf tc + jxQ (t)e j2πfct ]

x(t)= Re[xI (t)[cos(2πf tc )+ jsin(2πf tc )]+ jxQ (t)[cos(2πf tc )+ jsin(2πf tc )]]

------------- (3.14) x(t)= xI (t)cos(2πf tc )− xQ sin(2πf tc )

where is “in-phase component” and is “quadrature component” of the band xI (t) xQ (t) pass signal x(t). This nomenclature recognizes that sin(2πf tc ) is in phase-quadrature with respect to cos(2πf tc ).

Both and are low-pass signals limited to the bandxI (t) xQ(t) − w < f < w.

Hence, except for scaling factors, they may be derived from the band pass signal x(t) using the block diagram shown 3.7.

Figure 3.7: Block diagram to produce in-phase and quadrature components