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

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3
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1answer
81 views

Does analytic closed form solution exist for this trigonometric equation?

solve for x: $\sin(ax)=k\sin(bx)$, a,b,k and x are real numbers I am looking for a very general solution when a,b and k are completely unrelated.
1
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2answers
49 views

What is wrong with this basic algebra/trig that I'm doing?

I have to solve for $x$: $e^{x\sqrt3}(3cos(3x)+\sqrt3sin(3x))=0$ So $3cos3x+\sqrt3sin3x=0$ Divide through by$cos3x$: $3+\sqrt3tan3x=0$ $tan3x=-\dfrac{3}{\sqrt3}$ $\therefore$ ...
3
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3answers
377 views

2014 AMC 12 B problem 25

What is the sum of all positive real solutions $x$ to the following equation? $$2\cos(2x)\left( \cos(2x) - \cos{\left(\frac{2014\pi^2}{x^2}\right)} \right) = \cos(4x) - 1 $$
0
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1answer
35 views

$2\sin2\theta=\cos\theta$ from $0^\circ$ to $360^\circ$

Right, I took two different approaches and I don't get all of the correct answers from the second approach. I can't seem to figure out what is wrong with the second approach, a friend suggested that ...
2
votes
4answers
89 views

How to solve this limit of a function? ($\cos^3x$)

So I'm having trouble with the following limit: $$\lim_{x\to0}{\frac{1-\cos^3x}{x\,\sin x}}$$ Sorry to bother again, but I was never good at solving limits. Really, I don't know what to do with ...
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3answers
59 views

How to solve this limit of a function?

So I'm having trouble with the following limit: $$\lim_{x\to0}{\frac{x\sin x}{1-\cos2x}}$$ Tried to solve it multiple times and failed, so i posted it here... If possible, solve it in steps, so I ...
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vote
1answer
71 views

Integration of exponential trig functions

If $\cos^2x=[1+\cos(2x)]/2=(1/2)[1+\cos(2x)]$ Would I be wrong in assuming that $$\cos^2(3x+1)=\frac{1}{2}\left[1+\cos[2(3x+1)]\right]=\frac{1}{2}\cos(6x+2)+\frac{1}{2}?$$ I'm trying to take the ...
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1answer
289 views

Trig Problem. Find the angle.

I've spent hours on this problem now and I can't figure it out. :( I would very much appreciate some help. Thank you! A soccer field has a rectangular penalty area that measures 136 feet by 51 feet. ...
0
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1answer
34 views

Location on circle where tangent intersects a point in space

Given a point in space (x, y), and a circle with a radius (r) centered at (a, b), how would I calculate the point on the diameter of the circle where the tangent would pass through point (x, y)?
3
votes
1answer
457 views

Arc Length from chord and tangent angle

This is for a rubberband-powered car competition. In the diagram above, I will be given the length from points A to B, as well as angle a. The car will need to go from A to B, positioned at a to ...
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1answer
66 views

Prove this trigonometry inequality

I'm having difficulty proving that tan(26°) < 0.5 < tan(27°) . Any idea ? Thanks. p.s. 26 and 27 are in degrees.
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1answer
43 views

How show this $\Delta ABC\sim\Delta A'B'C'$

Assmue that:if $\Delta ABC$ and $\Delta A'B'C'$ are not right triangle,and such $$\sin{(2A)}:\sin{(2A')}=\sin{(2B)}:\sin{(2B')}=\sin{(2C)}:\sin{(2C')}$$ show that $$\Delta ABC\sim\Delta A'B'C'$$ ...
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vote
2answers
67 views

$2\sin x=3\cot x$ from $0^\circ$ to $360^\circ$

I used two slightly different approaches to solve this. First approach gives 2 correct solutions, second approach gives 4 solutions of which 2 are correct and 2 wrong, I just cannot figure out why ...
3
votes
1answer
114 views

Which one is greater, $\sin(\sin(\sin(1)))$ or $\cos(\cos(\cos(1)))$

I know I asked a similar question sometime before, and the thing is I need them for a proof. So, please help, I promise this is the last one. What is the simplest way we can find which one of ...
2
votes
2answers
116 views

Which is greater, $\cos(\cos(1))$ or $\cos(\cos(\cos(1)))$?

What is the simplest way we can find which one of $\cos(\cos(1))$ and $\cos(\cos(\cos(1)))$ [in radians] is greater without using a calculator [pen and paper approach]? I thought of using some ...
3
votes
3answers
294 views

$\arcsin(\sin x)$ explanation?

First off, I know this is a duplicate of this question. I'm asking this because I still don't quite understand the answer given there. But first, some graphs! $y=\sin(x)$ $y=\arcsin(x)$ ...
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3answers
88 views

Integral of $\int^\sqrt2_1\frac{1}{1+\sqrt{x^2 - 1}}dx$ by substitution?

In a maths question I have the question: $$\int^\sqrt2_1\frac{1}{1+\sqrt{x^2 - 1}}dx$$ by substitution? All other questions have been by trigonometric substitution so I assume that is how to solve. ...
19
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2answers
1k views

Tough contest problem

I found this problem in a collection of contest problems of a Russian competition in 1995 and wasn't able to solve it. Solve for real $x$: $$ \cos (\cos (\cos (\cos(x))))=\sin (\sin (\sin (\sin ...
0
votes
1answer
94 views

Prove trigonometric identity for $\tan^2(\theta/2)$

I've been at this for a couple of days (3-4 hours total now), and am feeling lost: $$(1)\ \tan^2(\frac{1}{2}\theta) = \frac{\tan(\theta) - \sin(\theta)}{\tan(\theta)+\sin(\theta)}$$ I'm aware that ...
2
votes
1answer
58 views

finding maxima of a function

I want to characterize all (countably infinite) maxima of this function $f(x) = \frac{\text{sin}^{2}(x)}{x}$. I tried the derivative approach, and it gives me the equation: $\frac{\text{tan}(x)}{x} = ...
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vote
1answer
42 views

Simplify this expression?

I have the following expression $$\frac 12 x_0e^{-\beta t}\left[\left(\frac {\beta}{i \sqrt{\omega ^2-\beta ^2}}+1\right)e^{i \sqrt{\omega ^2 - \beta ^2}t}+\left(\frac {- \beta}{i \sqrt{\omega ^2 - ...
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vote
0answers
96 views

The distance between two distinct points in the upper half plane

I'm trying to derive the distance between two distinct points in hyperbolic space and I'm working on the upper half plane. So, with the parametrization $\sigma(t): x=r\cos(t), y=r\sin(t),\; ...
0
votes
1answer
37 views

Trigonometry- cos and sec

Is $\cos^2 (x)$ the same as $1/\sec ^2 (x)$ ? I wasn't sure because $\cos$ is the same as $1/\sec$. thank you!
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2answers
111 views

Trigonometric Series Proof

I am posed with the following question: Prove that for even powers of $\sin$: $$ \int_0^{\pi/2} \sin^{2n}(x) dx = \dfrac{1 \cdot 3 \cdot 5\cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots 2n} \times ...
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3answers
112 views

Trigonometry, knowing 3 sides how to find the height?

I have a mathematician problem where, I knew the 3 sides of a triangle, with these sides I can figer out what type of type of triangle is. What I realy want to find is the height of the triangle and ...
0
votes
1answer
49 views

Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$

I'd like to calculate the following integral on the interval $[0,2\pi]$: $$ I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi. $$
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3answers
4k views

Using De Moivre's Theorem to prove $\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$ trig identity

I am stuck on trying to prove a trig identity using De Moivre's theorem. I have to prove, $$\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$$ I am not sure where to even start, I broke the LHS down ...
2
votes
2answers
83 views

Did I solve this integral correctly? (trig substitution)

I'm having trouble with trig substitution. This is what I've done so far, but I'm not sure if I did everything right. This is the integral: $$\int \frac{x^2}{(1+x^2)^\frac{3}{2}}$$ and my ...
2
votes
2answers
58 views

Trigonometric functions and formulas

Given that $\csc(\theta) = −\sqrt{2}$, $\tan{\theta} = 1$ and $−\pi < θ < \pi$, find the exact value of the angle $\theta$ in radians.
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votes
2answers
439 views

$\tan(\pi/2)$ is undefined but $\cot(\pi/2)$ is defined

We know that $\tan(\pi/2)$ is undefined, and $\cot(\pi/2) = 0$. But $\cot(x) = \frac{\cos(x)}{\sin(x)} = \frac{1}{\tan(x)}$. So how is it possible that for some value $x$, $\tan(x)$ is undefined but ...
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8answers
248 views

Let $\theta=\frac{2\pi}{5}$. Show $2\cos(2\theta)+2\cos(\theta)+1=0$.

I have been at this for a while. Any ideas?
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1answer
24 views

Concept of parallelism in analytic terms

Below I cited a passage from Apostol's Calculus. I don't understand how to use the identity to show that two lines with equal slopes are parallel. Concepts such as perpendicularity and parallelism ...
3
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0answers
286 views

Trigonometric functions of angle fractions

I've just encountered a problem that seems to me interesting enough so that some result exists on the subject. I was working on a problem in complex analysis, in which I needed the fifth root of a ...
0
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1answer
118 views

Generating function of trigonometric fuction

Find exponential generating function for following sequence: $s_{n} = \sin{nt}$ the answer should be in terms of trigonometric functions. The exponential generating function is defined as: $S(x)= ...
2
votes
1answer
132 views

Definite integral involving arctan and tan

I was solving a problem posed on Moldavian National Mathematical Olympiad for 12th grade in 2012. The question was the following: Problem. Let $f:\mathbb{R}\rightarrow\mathbb{R}$, such that ...
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2answers
471 views

How to solve this trig problem? $\sec(\sin^{-1}(-5/13)-\tan^{-1}(4/3))$

Basic trig problem my brother ask me, but I don't know how to do it: $$\sec(\sin^{-1}(-5/13)-\tan^{-1}(4/3))$$
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3answers
38 views

Series representation of $\sin(nu)$ when $n$ is an odd integer?

So, out of boredom and curiosity, today I came up with a series representation for $\sin(nu)$ when $n$ is an even integer: $$\sin(nu) = \sum_{k=1}^\frac n2 ...
10
votes
2answers
209 views

Proof that cos(1) is transcendental?

So, I was playing around on Wolfram|Alpha (as we nerds like to do) and it said cos(1) was transcendental. Could someone provide me with the proof that cos(1) is transcendental?
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3answers
120 views

How can this trig equation be simplified?

We have $9+40\sin^2x=-42\sin x\cos x$. I know this simplifies to $7\sin x+3\cos x=0$, but how?
2
votes
1answer
294 views

Conclusion from trigonometric identity

Let $\alpha$ and $\beta$ be angles in triangle, i.e $\alpha, \beta \in \left(0,\pi\right)$ can we conclude that $\alpha = \beta$ if the following statement is true: $$\left(\frac{\sin \alpha}{\sin ...
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2answers
267 views

Why does $\sin^{-1}(\sin(\pi))$ not equal $\pi$

And when does $\sin^{-1}(\sin(x)) = x$
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5answers
456 views

Is this trig step correct?

$\sin^{-1}(-\sin(x))$ = $-\sin^{-1}(\sin(x))$ Can the minus be taken out like this?
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2answers
83 views

trigonometric equation for tan^(-1)

Need to solve this equation: $$\tan^{-1}\frac{x}{3} +\tan^{-1}x= \tan^{-1}2.$$ Method with explanation is highly appreciated
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3answers
153 views

How to calculate $\theta$ when we know $\tan \theta$.

Hej I'm having difficulties calculating the angle given the tangent. Example: In a homework assignement I'm to express a complex variable $z = \sqrt{3} -i$ in polar form. I know how to solve this ...
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2answers
138 views

Finding the length of the opposite and adjacent sides of a triangle

I am writing a small game in javascript. It's been a while since I have done any basic maths and I can't get some of my positioning to work properly. Apologies if this question is too simple, but I ...
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5answers
278 views

Simple trigonometric identity proof

How would you verify that this trigonometric equation is an identity? $$\sin^4x-\cos^4x=2\sin^2x-1? $$ The 4th powers are really throwing me off, and i'm still fairly new to this and there is no ...
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3answers
43 views

Simple Trigonometry and algebra

If $$\sec\theta = X + \frac{1}{4X},$$ then what is $${\sec\theta + \tan\theta}$$ in terms of $X$?
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2answers
73 views

How to solve this simple trignometric problem?

So this is the question that was given in a textbook and i attempted to win from the book which was saying i was wrong? If $$\frac{\sin\theta + \cos\theta}{\sin\theta - \cos\theta} = ...
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votes
2answers
3k views

What exactly do the sin, cos, tan buttons do on a calculator?

I understand they mean sine, cosine, tangent, but what exactly is the calculator doing when I enter an angle and press those buttons? Edit: To help others better understand my question, my question ...
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2answers
218 views

How is the sin function being rewritten?

I'm working through a trigonometry book and was shown this equation being worked out. I don't understand the rules for doing a particular step: $$\begin{align} A &= A\sin(x-vt) \\ 1 &= ...