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

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1answer
45 views

Solving trigonometric equation for $a$ and $b$

Does anybody know of a quick way of finding real $a$ and $b$ for the equation $2\sin (x+10)= a\sin x + b\cos x$?
6
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2answers
750 views

Method to find $\sin (2\pi/7)$

I just thought a way to find $\sin\frac{2π}{7}$. Considering the equation $x^7=1$ $⇒(x-1)(x^6+x^5+x^4+x^3+x^2+x+1)=0$ $⇒(x-1)[(x+\frac1 x)^3+(x+\frac1 x)^2-2(x+\frac1 x)-1]=0$ We can then get the ...
4
votes
1answer
111 views

Total variation of a Fourier series

Let $f(x) = f(x+2\pi)$ be a bounded real function given by the Fourier series of the form $$ f(x) = \sum_{k=1}^N a_k \sin(kx + \phi_k). $$ What is the total variation $V(f)$ of this function over one ...
2
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1answer
247 views

How to calculate the position of a turning object, based on its rotation?

I'm working on a program that periodically updates the position of an object. The object is able to move in straight lines, as well as turn gradually. In order to check that my object is turning ...
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2answers
396 views

Trigonometry: Proving Question involving Sum to Product

I have a homework question that asks me to prove the following: $$\frac{\sin \theta+\sin 7\theta}{\sin 3\theta+\sin 5\theta}=2\cos2\theta-1$$ When I tried proving it, I could only do $$LHS=\frac{\sin ...
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9answers
6k views

Good book recommendations on trigonometry

I need to find a good book on trigonometry, I was using trigonometry demystified but I got sad when I read this line: Now that you know how the circular functions are defined, you might wonder how ...
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2answers
108 views

Central limit theorem using it with formulas

1.The average amount of money spent at lunch in the wissakion cafeteria is 3.00 dollars with a standard deviation of 75 cents. Assume the distribution of money spent is normal. a. What is the ...
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2answers
65 views

Trigonometric proof query

I am having trouble proving the following identity (where $m,n \in \mathbb{R}$ are arbitrary): $$\sin(mx)\sin(nx) = \frac{1}{2}[\cos(m -n )x - \cos(m + n)x] \quad (1)$$ By expanding the RHS, I can ...
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1answer
75 views

I need to trace the edge of a hexagon point by point

To create an animation in code I first need to be able to describe a point by point path around a hexagonal shape. I already have the $\displaystyle \frac{x}{y}$ coordinate of each vertex. I don't ...
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1answer
331 views

how to prove $\displaystyle \frac{\sin (2n+1)\theta}{\sin \theta} = … $

How to prove $$ \displaystyle \frac{\sin (2n+1)\theta}{\sin \theta} = (2n+1) \prod_{k=1}^{n}\left(1 - \frac{\sin^2 \theta}{\sin^2 \left( \frac{k\pi }{2n+1} \right ) } \right ) $$ So far, I manage to ...
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0answers
503 views

On Ramanujan's Question 359

In JIMS 4, p.78, Question 359 was asked by Ramanujan. If, $\sin(x+y) = 2\sin\big(\tfrac{1}{2}(x-y)\big)\tag1$ $\sin(y+z) = 2\sin\big(\tfrac{1}{2}(y-z)\big)\tag2$ prove that, $\big(\tfrac{1}{2}\sin ...
4
votes
1answer
126 views

Polynomial expression of $\frac{\sin x}{x} $

Could you explain to me why $$\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi ^2}\right)\left(1-\frac{x^2}{(2 \pi) ^2}\right)\left(1-\frac{x^2}{(3 \pi )^2}\right)\cdots$$ I've read in this article ...
4
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2answers
190 views

proving $\csc^2 \left( \frac{\pi}{7}\right)+\csc^2 \left( \frac{2\pi}{7}\right)+\csc^2 \left( \frac{4\pi}{7}\right)=8$

How can I prove the following identity using complex variables $$ \begin{align*} 1) & \csc^2 \left( \frac{\pi}{7}\right)+\csc^2 \left( \frac{2\pi}{7}\right)+\csc^2 \left( \frac{4\pi}{7}\right)=8 ...
4
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1answer
111 views

How to solve this nonstandard trigonometric equation?

I want to solve this equation $$\sin(\sin(\sin(\sin x)))=\cos(\cos(\cos(\cos x))),$$ but I don't know how to solve.
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0answers
116 views

A construction of the trig functions on the unit circle

Can anyone shed some light on this picture? http://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg I am not interested in "$\sin$", "$\cos$", or the outdated trig functions. How do we ...
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1answer
86 views

Sum of complex numbers $z_k=\exp{\Big(ik\varphi\Big)}$

Show that $$\sum_{k=-j}^jz_k=\sum_k\exp{\Big(ik\varphi\Big)}=\frac{\sin{\varphi(j+\frac{1}{2})}}{\sin{\frac{\varphi}{2}}}.$$ I see that $\sum_kz_k=1+2\sum_{k=1}^j\cos{k\varphi}$. EDIT i write my ...
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1answer
82 views

Maximum value of the given expression

Assuming $\theta \in [\frac{-5\pi}{12},\frac{-\pi}{3}]$, find the maximum value of $$\frac{\tan(\theta+\frac{2\pi}{3})-\tan(\theta+\frac{\pi}{6})+\cos(\theta+\frac{\pi}{6})}{\sqrt{3}}$$ I know we can ...
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3answers
187 views

Verify that: $2\cot{x}/\tan{2}x = \csc^2x-2$

Verify the following: $$\frac{2\cot{x}}{\tan{2}x} = \csc^2x-2\;.$$
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4answers
487 views

Verifying Trigonometric Identities: $2\cos^2x-1 = \frac{1-\tan^2x}{1+\tan^2x}$

Verify that $$ 2\cos^2x-1 = \frac{1-\tan^2x}{1+\tan^2x}$$
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3answers
969 views

Calculate the $\int_0^{2\pi}\cos(mx)\cos(nx)dx$

I'm having trouble with this problem: Consider the integral: $$\tag 1\int_0^{2\pi}\cos(mx)\cos(nx)dx$$ a. Write $\cos(mx)$ and $\cos(nx)$ in terms of complex exponentials and compute ...
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1answer
54 views

Solving a trig equation

How would I solve the following trig equation $\sin^2x=1-\cos(x)$ I have to write the solution in radians.
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3answers
294 views

Simplification of $\sqrt{(1-\cos\alpha \cos\beta)^2-\sin^2\alpha \sin^2\beta}$

Simplify the expression $$\sqrt{(1-\cos\alpha \cos\beta)^2-\sin^2\alpha \sin^2\beta}$$ I have done this way : $(1-\cos\alpha \cos\beta)^2 = 1-2\cos\alpha \cos\beta +\cos^2\alpha \cos^2\beta$ ...
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2answers
132 views

Are there rigorous mathematical definitions for these waves?

My friend linked this .gif to me tonight, and asked me if I knew of any equations that might model these bottom two waves (the blue and green waves). Unfortunately, I am not far enough in my education ...
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2answers
68 views

Solve Trignometric Equation

Solve the equation: $\sin^25x+\sin^23x = 1+\cos(8x)$. I tried : $1+\cos(8x) = 2\cos^2(4x)$ which gives : $$\begin{align*} \sin^25x+\sin^23x &= 2\cos^2(4x)\\ &= 2(1-\sin^2(4x))\\ ...
13
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4answers
613 views

Showing that $ |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$

For every nonnegative integer $n$ and every real number $ x$ prove the inequality: $$\sum_{k=0}^n|\cos(2^kx)|= |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$$
26
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7answers
2k views

Where are the values of the sine function coming from?

On high school, I was taught that I could obtain any sine value with some basic arithmetic on the values of the following image: But I never really understood where these values where coming from, ...
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1answer
61 views

How to solve given trigonometric equation

I've got a stream function: $$u_\infty y + \frac{Q}{2\pi} \operatorname{arctg}\frac{y}{x} = 0 $$ How do I solve it for y? I know the solution, just don't know how to get there step by step. EDIT: ...
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2answers
121 views

Integral of a weird trigonometric function

I'm been trying to figure this out for hours, but no success. Can anyone take a look at it? Thanks a lot! $$\int\frac{1}{\sin2x + \cos2x}dx\qquad\text{Hint: start by evaluating }\int\frac{1}{\sin ...
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3answers
409 views

Find the exact value of $\cos[\arcsin(-\frac{1}{3})]$

This is for my homework and I don't know how to approach it to get an exact value. Given that $- \frac{\pi}{2} \le \arcsin x \le \frac{\pi}{2}$, find the exact of $\cos[\arcsin(-\frac{1}{3})]$.
5
votes
2answers
330 views

Can the distance from the vertices of a square of integer width to an inscribed circle all be integer?

I'm looking for solutions to the following British Mathematical Olympiad question: Suppose that $ABCD$ is a square and that $P$ is a point which is on the circle inscribed in the square. Determine ...
3
votes
1answer
97 views

$X=(1 + \tan 1^{\circ})(1 + \tan 2^{\circ})(1 + \tan 3^{\circ})\ldots(1 + \tan {45}^{\circ})$. what is the value of X? [duplicate]

$$X=(1 + \tan 1^{\circ})(1 + \tan 2^{\circ})(1 + \tan 3^{\circ})\ldots(1 + \tan {45}^{\circ})$$ $$\tan(90-\theta)=\cot\theta=\frac{1}{\tan\theta}$$
5
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2answers
4k views

Are $\mathrm{arccot}(x)$ and $\arctan(1/x)$ the same function?

In my textbook it asks for me to: Prove that there is no constant $C$ such that $\text{arccot}(x) - \text{arctan}(\frac{1}{x}) = C $ for all $x \ne 0$. Explain why this does not violate the ...
3
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1answer
404 views

How are the parametric equations describing the cupid curve derived? [duplicate]

No doubt as some people have already seen, today morning wolfram posted the best valentine ever. The graph depicting cupid with its arrow and floating hearts around it involves something like 6 pages ...
4
votes
1answer
123 views

Is it allowed to divide an equation by an expression which can be equal to zero?

I need a help in such a problem and will greatly appreciate any suggestions. I was taught, division of an equation by an expression which can be equal to zero can lead to missing roots. But I thought ...
2
votes
1answer
102 views

How does this periodic trig function that calculates modulus work?

$$ \arctan(\tan(( \mathrm{dividend} - \frac{\mathrm{divisor}}{2}) \times \frac{\pi}{\mathrm{divisor}}))*\frac{\mathrm{divisor}}{\pi}+\frac{\mathrm{divisor}}{2}= \mathrm{dividend} \bmod ...
3
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2answers
140 views

Different solutions of trigonometric equations

Please take a look at this trigonometric equation, $\cos9x\cos7x = \cos5x\cos3x$ To solve this equation, we can proceed as, $2\cos9x\cos7x = 2\cos5x\cos3x$ or, $\cos(9x+7x)+\cos(9x-7x) = ...
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2answers
503 views

How do I use the appropriate addition formula to find the exact value of this expression

Use the appropriate addition formula to find the exact value of the expression. $\sin\left(\large\frac{11}{12}\pi\right)$
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4answers
364 views

loss-of-significance error

Reduce loss of significance error in the following equation by re-arranging terms: $f_1(x) = \frac{1- \cos(x)}{x^2}$ , assuming $x$ is near $0$. Let $f_2(x)$ be the function rewritten to reduce loss ...
3
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1answer
80 views

Solve this Trigonometric equation

I am not quite good at maths, so can you help me ? $$\tan x + 2 \cot x - 3=0$$
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3answers
1k views

Finding all solutions to $ \tan^5x - 9\tan{x} = 0 $

I am stuck when it comes to finding the end value of a trig function. I have the following question: $$ \tan^5x - 9\tan{x} = 0 $$ I worked the problem and got: $$ \tan x = 0\\ \tan^4x-9 = 0\\ x = ...
3
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7answers
19k views

Calculate the height of a distant object using estimated angles from two different points.

I knew how to do this long ago, found the exact problem in my old trig book, but I can't seem to work it out. Say I'm at an unknown distance from a mountain, called point P, and I estimate the angle ...
6
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3answers
948 views

Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$

I'm tutoring for a college math class and we're doing putnam problems next week and this one stumped me: Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers ...
3
votes
2answers
470 views

Funny Trig Math Puzzle

This is a challenging puzzle I heard from my little brother. For some $n$ and $x$, $\sum_{k=1}^n \sin^{2k}(x) = 2013$. Is it possible to deduce $$\sum_{k=1}^n \cos^{2k}(x) \text{ ?}$$ Edit: I've ...
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5answers
245 views

Prove the identity $1 + \sin x = 2 \cos^2 \left(45° - \frac{x}{2}\right)$

Here is the problem: $$1 + \sin x = 2 \cos^2 \left(45° - \frac{x}{2}\right)$$ Can you help me prove that this is an trigonometric identity?
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1answer
135 views

Another trigonometric equation

Show that : $$31+8\sqrt{15}=16(1+\cos 6^{\circ})(1+\cos 42^{\circ})(1+\cos 66^{\circ})(1-\cos 78^{\circ})$$
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0answers
131 views

Some trigonometric equation problems

show that : $$\left(1+\cos \frac{2\pi}{13}\right)\left(1-\cos \frac{4\pi}{13}\right)\left(1+\cos \frac{6\pi}{13}\right)\left(1+\cos \frac{8\pi}{13}\right)\left(1-\cos ...
0
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3answers
928 views

Simplifying trigonometric functions

I am having a lot of problems simplifying the following trig functions, please help: Simplify: $\sec x\tan x(1 - \sin^2 x)$. Thank you Tony
4
votes
4answers
919 views

Does $\sin^2 x - \cos^2 x = 1-2\cos^2 x$?

I am finishing a proof. It seems like I can use $\cos^2 + \sin^2 = 1$ to figure this out, but I just can't see how it works. So I've got two questions. Does $\sin^2 x - \cos^2 x = 1-2\cos^2 x$? ...
2
votes
0answers
262 views

Desired Z axis and Yaw to ZXY Euler Angles?

I'm trying to calculate a desired pair of pitch and roll Euler angles (the XY in ZXY format) given a desired z-axis of the rotated frame (expressed in the world frame) and a specified yaw angle ...
-1
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3answers
232 views

matrix representation of a trigonometric rotation

Hey guys!I have a couple of doubts regarding this exercise, for a) I think that the Matrix rotation of P is [(cos t, -sen t) , (-sen t, cos t)] and for Q [(-cos t, -sen t), ( sen t, cos t)] , is ...