Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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24 views

$\int_0^\pi \cos{mx}\cos{nx}\,\mathrm{d}x$, with $m$ and $n$ integers

I can't figure out this integral at all. Integrating gave the following \begin{equation} \frac{1}{2}[\frac{\sin((m-n)x)}{m-n}+\frac{\sin((m+n)x)}{m+n}], \end{equation} but when I evaluate it from ...
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38 views

how to plot the graphs of $\sin^{-1}(\sin x)$,$\cos^{-1}(\cos x)$,$\tan^{-1}(\tan x)$

Can someone please explain me how to plot the graphs of $\sin^{-1}(\sin x)$,$\cos^{-1}(\cos x)$,$\tan^{-1}(\tan x)$ ? I am having a little difficulty in understanding how the nature of the graphs can ...
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37 views

Find reflected angle off an arc

Note: Sorry if this question is elementary. I am trying to find the reflected angle when a ball hits a 10 degree arc. I know I have to use trigonometry in some way, but I am having trouble grasping ...
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92 views

Squeeze theorem and $\frac{\sin x}{x}$

I've been going over old calculus books to refresh my memory and have mainly been focusing on proofs. One of the things I found interesting was the squeeze theorem, even though since basic calculus i ...
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10 views

trigonometry - find coordinates of inner triangle after rotation

here is my situation: I have a rectangle I'm rotating 30 degrees counterclockwise, how could I use trig to get the 3 vertices (corners) and lengths of the purple triangle sides and hypotenuse ...
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34 views

Values of $\arctan(x)$ by hand

How can I solve arctan problems by hand? Is this a matter of hoping for good forms and memorizing a long list of them? My question had as part of it: $$z=1-\sqrt3i$$ and I can immediately see that ...
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43 views

Why is the argument of complex number determined up to integer multiple of $2 \pi$?

I have just started learning about complex number and came across to this argument of complex number Let's say we have a complex number $z$. Then the argument of $z$ can be represented by this: $arg ...
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33 views

How can you quickly ease a sine wave's amplitude?

I'm currently drawing a sine wave using the following function in java: Math.sin(x)*currentAmplitude The problem I'm experiencing is that, although this works ...
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41 views

How can I derive Polarization Operator algebraically

Can anyone describe for me the algebraic steps needed to create what Prof. Leonard Susskind calls, in one of his early Quantum Mechanics courses, the "Polarization Operator Matrix"? It is a 2 by 2 ...
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32 views

How to calculate Arc Cosine, without a unit circle.

How can I calculate the $\arccos(Y)$, neither using the trig functions on a calculator, nor using a unit circle? I would be satisfied if I got $\arcsin(Y)$ or $\arctan(Y)$ instead, but I would prefer ...
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26 views

closed form or approximate solution for a system of equation : $m(t)=v\sin(\arctan(at+b))+v\sin(\arctan(ct+d))$

Can one solve for $(v,a,b,c,d)$ the following equation ? $t$ takes discrete values and $m(t)$ is known for as many $t$ needed. However please assume that special values of $t$ may not be available ...
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16 views

Workout line segment inside expanding circle

I have what is probably a fairly basic math problem for a game I'm creating. On each frame I need to work out how much a sub segment of a line passing though a circle will expand when the circle ...
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6 views

Solutions of $\sum_{n=1}^N a_n n\sin{(n x+\theta_n)}=\sum_{n=1}^N a_n n^2\cos{(n x+\theta_n)}=0$

Is there a solution for the equation $\sum_{n=1}^N a_n n\sin{(n x+\theta_n)}=\sum_{n=1}^N a_n n^2\cos{(n x+\theta_n)}=0$ in terms of the variable $x$, for some choice of coefficients $a_n$ and ...
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34 views

What is the expansion of $(a\sin A +b\sin B)^n$ in terms of $\sin$ or $\cos$ by power of zero and one?

What is the expansion of: $(a\sin A+b\sin B)^n$ in terms of $\sin$ or $\cos$ by power of zero or one? For example the second expansion is: $(a\sin A+b\sin B)^2=$ $a^2+b^2+$ ...
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61 views

Rewriting $\sin 2B$ in terms of sine and cosine functions

I am not a math person. I am wondering if $\frac12 \sin 2B$ will resolve to $(\sin B \cos (B + Q)) / \cos Q$. Sorry, first time posting here as well as having to learn latex a little, I ...
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17 views

trigonometric expressions as algebraic numbers

which trigonometric expressions give results which are rational or expressible as surds? is there a complete set? are there infinite? for example, the well known $sin(30)=1/2$, and a range of others ...
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18 views

Sinusoid with zero boundary conditions on $[0,1] \times [0,1]$ grid

I want to make a sinusoidal plot (any shape is welcome) on a $[0,1] \times [0,1]$ grid, with boundary conditions equal to zero. It should resemble a membrane fixed along its edge. I tried out some ...
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20 views

What is the name for representing a polygon as arcs on a circle?

I am having trouble finding the name of a particular concept. I did a bit of work with it about 10 years ago >__>. The idea went something like this-- Any polygon may be represented or expressed as a ...
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25 views

Write $A\cos\theta_1 + \cos\theta_2$ as product of cosines

I would like to write $A\cos\theta_1 + \cos\theta_2$ as product of cosines. If $A=1$ and $x=\operatorname{Re} z = \cos\theta_1 + \cos\theta_2$, where $z=e^{i\theta_1}+e^{i\theta_2}$. Then ...
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52 views

Historical calculations of $tan^{-1}x $ and $e^x$

SineBhaskara_I One reads that $tan^{-1}(x) $ series expansion existed in early (Indian) history. But like the Sine trigonometric function, did any similar approximation exist as well? The query ...
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29 views

Osborne's rule for hyperbolic functions?

I am confused as to why you only change the sign for powers of sine that are 4n+2. As I understand, $sin(i\theta)=isinh(\theta)$ $sin^2(i\theta)=-sinh^2(\theta)$ $sin^3(i\theta)=-isinh^3(\theta)$ ...
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31 views

Trigonometry - word problem - functions cosθ = adjacent/hypotenuse

I'm not sure what trigonometric equation I should use for this problem: At a certain instant, a ship was 5km south of a lighthouse. The ship was travelling westward and after 30 minutes it's bearing ...
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14 views

Tricky Harmonic Progression

If the numbers 1+x,1-y,x,y is a sequence of numbers in a harmonic progression, show that all the different values of x+y are given by cos[2*(3*n+1)*pi/9] for n = 0,1,2
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25 views

Pre Calculus Simplifying Trigononmetric Expressions

I am trying to simplify this trigonometric expression in terms of sine and cosine. The equation that I have to simplify is 7 cos t tan t.
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33 views

Constructing a pentagon from a circle

To my understanding you can create any regular N sided shape by using a circle. I decided to give this an attempt from the equations/ formulas given from the internet. Just a side note - is it ...
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41 views

Legendre Polynomials Integration

I have the original equation as follows: x_jj'= Integral of Pj(cos(x))*cos(x)*Pj'(cos(x))*sin(x)dx from 0 to pi where Pj is the Legendre Polynomial with the j ...
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18 views

Trigonometric functions: same function values

What's a good method to determine at what point two (or more) trigonometric functions give map to the same point for the same two (or more) $x$? E.g. we have $\sin{x}$ and $\cos{x}$ that map to the ...
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14 views

parametrization: t values that lead to the same point

Given a parametrization $r(t)$, we can have several $t$ values that correspond to the same point on the curve... what's the method of finding all of these when dealing with trigonometric functions? I ...
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33 views

Mapping cartesian coordinates to circular coordinates

I have some "rectangular" x,y coordinates which I like to convert to "circular" coordinates. Notice I'm not sure if I'm using the correct terms here, so bear with me - I drew an example to show how ...
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35 views

Proofing an equation using a figure

I was given the following assignment by my tutor, but I cannot figure it out. I'm hoping you guys have any clue on how to proof the equation. The following figure was given: I need to answer the ...
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32 views

Analytical solution to equation $ \arctan(x)-k \arctan(x/k)=c$

For the equation:$$\arctan(x)-k \arctan(x/k)=c$$ which is part of a gasdynamics function called Prandtl–Meyer function, it is not difficult to find the solution numerically, however, I'm wondering, ...
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69 views

What does $-\pi < \theta < \pi$ mean?

When I was asked to provide an angle in the interval $-\pi < \theta < \pi$, I said $\frac{3\pi}{4}$ since this is obviously less than $\pi$, but apparently it is not greater than $-\pi$ ...
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17 views

Formula for the base edges of a rotated cone

I need to create a set of equations to find points along the edge of the base of a cone, but I'm stuck What I have: The cone can be rotated over any plane in $\mathbb{R}^3$. The position of the ...
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30 views

Trigonometric Question

How can I convert numerical decimal values such as $1.0$ , $0.71$ or $0.53$ into degrees? For example, $1=\sin90^\circ$ $0.71=\sin?$ (I really need help) Is there any equation to convert the ...
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44 views

What does this trigonometric function represent?

I'm doing an investigation on this trig function and can't seem to find what it relates to as far as real world uses. Any help greatly appreciated. $f(x) = cos ({k\over x^2+1})$ Edit: Basically, ...
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62 views

If $\cos(A + B) < 0$, then why $-\pi \leq A + B \leq -\frac{\pi}{2}$?

The value of $A + B$ should be less than $\frac{-\pi}{2}$ for the cosine to be negative. Then relation should be $-\pi \leq A + B < -\frac{\pi}{2}$. But the question's relation is not this; that ...
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59 views

New Scientist Enigma 1779

I drew four right-angled triangles. The hypotenuse of my first triangle was also the shortest side of my second triangle; the hypotenuse of my second triangle was also the shortest side of my third ...
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30 views

Angle of Sine wave

How you do calculate angle of sine wave? Here in this example you can see the angle as the sine wave goes either side of the graph http://www.mathopenref.com/triggraphsine.html. For producing the sine ...
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27 views

Find Z component of a 3-dimensional vector's magnitude

So, I'm quite confused. I'm currently working with 3-dimensional vectors in an attempt to model an object's local axes in Unity. I already understand how to find the X and Y components of said ...
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22 views

Convex Constraint on Sine Wave Simularity

So lets say you have a vector X = [x1 x2 x3 ..... xn] You want to optimize a cost function over X. However you want to constrain the vector X to look like a sine wave. Say you can parameterize a ...
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34 views

Fraction of the area of multiple squares overlapped by a circle

This is a practical question based in a programming problem I have. Basically, I have a database full of latitude and longitude points, spaced out to be 1km from each other, corresponding to a ...
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32 views

Projecting point in 3d space onto a 2d view

If I have the following information: The coordinates in 3d space of a point(x, y, z) The dimensions of a 2d viewing window(width, height) The coordinates in 3d space of the center of that view(x, y, ...
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15 views

Sinusoid: go back from arbitrary point to p/4

If you pick an arbitrary point on a sinusoid, and you move to the left, is there a way to calculate where you have reached the point that is 1/4 the period of the sinusoid to the left of the start ...
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42 views

Find coordinates of a point 30degrees from another point

I need to find the coordinates of a line that is 30degrees away from another point. (If you look on the attached image it should explain, I want the coordinates of the top of all the blue lines.) I ...
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24 views

Trig in two Congruent Rectangles

I have two Congruent Rectangles (L by H), laid on top of each other, yet the second is rotated Θ˚around the center point. How Far away from the center line (dotted in diagram) is point X along the the ...
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60 views

Pythagorean identities

Pythagorean Identities:where $\sin^2x +\cos^2x=1$: when $x \geq 1$ the following examples are true. I don't know if this is original research or not but it states that for all integers bigger than ...
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21 views

Rotate vector around point - rotation matrix

x is the x-position of the point y is the y-position of the point i want rotate that point by $\alpha$ degree around the origin I found this formulas: $x = x⋅\cos \alpha−y⋅\sin \alpha$ $y = ...
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23 views

How to get angle with certain values, image provided

A, B and β are known, how to get value of α. I'm sorry I ask this easy question but after high school I was in army and I have forgotten even some basic algebra which I would now need for my coding ...
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14 views

Defining similarity through angle-preserving transformations

I'm hoping somebody can further explain the solution to this problem to me for me to understand. It provides two statements and asks if both, or one or the other are true.
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57 views

Calculate x,y,z given angles and magnitude of vectors

I am making a program where the user can input their desired velocity as well as pitch, yaw, and roll of an airplane, and then I will animate it. I am accomplishing this by updating it's position by ...