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

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17
votes
6answers
8k views

Exact values of $\cos(2\pi/7)$ and $\sin(2\pi/7)$

What are the exact values of $\cos(2\pi/7)$ and $\sin(2\pi/7)$ and how do I work it out? I know that $\cos(2\pi/7)$ and $\sin(2\pi/7)$ are the real and imaginary parts of $e^{2\pi i/7}$ but I am not ...
17
votes
4answers
4k views

How does e, or the exponential function, relate to rotation?

$e^{i \pi} = -1$. I get why this works from a sum-of-series perspective and from an integration perspective, as in I can evaluate the integrals and find this result. However, I don't understand it ...
10
votes
1answer
257 views

Proof of an equality involving cosine $\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{2}}}}\ =\ 2\cos (\pi/2^{n+1})$

so I stumbled upon this equation/formula, and I have no idea how to prove it. I don't know how should I approach it: $$ \sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{\vphantom{\large A}2\,}\,}\,}\,}\ ...
10
votes
2answers
216 views

Method of proof of $\sum\limits_{n=1}^{\infty}\tfrac{\coth n\pi}{n^7}=\tfrac{19}{56700}\pi^7$

The following formula was stated by Ramanujan: $$\sum\limits_{n=1}^{\infty}\frac{\coth n\pi}{n^7}=\frac{19\pi^7}{56700}$$ Does anybody know the method of proof of this formula? I know that typically ...
7
votes
5answers
17k views

Determine third point of triangle when two points and all sides are known?

Determine third point of triangle (on a 2D plane) when two points and all sides are known? A = (0,0) B = (5,0) C = (?, ?) AB = 5 BC = 4 AC = 3 Can someone ...
6
votes
4answers
1k views

Infinite series expansion of $\sin (x)$

Are there any other ways to demonstrate that $$\sin(x)=\sum_{k=0}^{\infty}\frac{(-1)^kx^{1+2k}}{(1+2k)!}$$ without using the definition of Taylor series of complex exponentials, and similarly for ...
8
votes
3answers
290 views

A trigonometric equation

Some years ago, I came across the following question: Find the value of $\tan5°\tan55°\tan65°\tan75°$. The value is 1 and I generalized the question as follows. $\tan a°\tan b°\tan c°\tan d° ...
6
votes
2answers
454 views

Reference for a tangent squared sum identity

Can anyone help me find a formal reference for the following identity about the summation of squared tangent function: $$ \sum_{k=1}^m\tan^2\frac{k\pi}{2m+1} = 2m^2+m,\quad m\in\mathbb{N}^+. $$ I ...
4
votes
7answers
364 views

Exact value for $\cos 36°$

Good morning! I'm having trouble with this problem... It's just taking me forever and I'm worn out and I'm lost on how to use a double angle identity for $72=2⋅36$ The problem reads as follows An ...
11
votes
5answers
3k views

Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?

(source for above graph) (source for above graph) Both functions simplify to x, but why aren't the graphs the same?
10
votes
2answers
5k views

What is the optimum angle of projection when throwing a stone off a cliff?

You are standing on a cliff at a height $h$ above the sea. You are capable of throwing a stone with velocity $v$ at any angle $a$ between horizontal and vertical. What is the value of $a$ when the ...
7
votes
3answers
1k 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 ...
7
votes
2answers
490 views

Prove that if $x$ is a non-zero rational number, then $\tan(x)$ is not a rational number and use this to prove that $\pi$ is not a rational number.

Prove that if $x$ is a non-zero rational number, then $\tan(x)$ is not a rational number and use this to prove that $\pi$ is not a rational number. I heard that this was proved two hundred years ...
6
votes
5answers
689 views

Picture/intuitive proof of $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$?

Is there a nice geometric, intuitive or picture proof as to why the easily algebraically provable identity $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$ is true? Note I'm not looking for a ...
6
votes
1answer
1k views

How to calculate a heading on the earths surface?

Given an initial position and a subsequent position, each given by latitude and longitude in the WGS-84 system. How do you determine the heading in degrees clockwise from true north of movement?
5
votes
4answers
13k views

Prove $\cos 3x =4\cos^3x-3\cos x$

How would I solve the following double angle identity. $$\cos 3x =4\cos^3x-3\cos x $$ I know $\,\cos 3x = \cos(2x+x)$ So know I have $\,\cos 2x +\cos x \,$ , Which is $\,(2\cos^2x-1)\cos x$ But ...
12
votes
4answers
1k views

For which angles we know the $\sin$ value algebraically (exact)?

For example: $\sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$ $\sin(18^\circ) = \frac{\sqrt{5}}{4} - \frac{1}{4}$ $\sin(30^\circ) = \frac{1}{2}$ $\sin(45^\circ) = \frac{1}{\sqrt{2}}$ ...
10
votes
2answers
915 views

How does $\cos(2\pi/257)$ look like in real radicals?

We know $\cos(2\pi/p)$ for p a Fermat prime can be expressed in real radicals. The case $p=17$ is a root of an 8th deg eqn, but can be also given as a sequence of nested radicals, $$\begin{aligned} ...
6
votes
3answers
475 views

Trigonometry Simplification

I have to simplify and evaluate this : $$\cot(70)+4\cos(70)$$ On evaluating it the answer comes out to be $1.732$ or $\sqrt 3$ . I tried to get everything in $\sin$ and $\cos$ but it doesn't go ...
6
votes
3answers
1k views

Partial Fractions Expansion of $\tanh(z)/z$

I have seen the following formula in papers (without citations) and in Mathematica's documentation about Tanh[]: $$ \frac{\tanh(z)}{8z}=\sum_{k=1}^{\infty} ...
5
votes
1answer
180 views

How do you plot the graph of $y = \sin(x)+\cos(x)$ by converting it to another form?

Are there any trigonometric identity that can make it easier to plot?? I have no idea how it becomes a sin graph shape in the end.
5
votes
3answers
7k views

How to find roots of $X^5 - 1$?

How to find roots of $X^5 - 1$? (Or any polynomial of that form where $X$ has an odd power.)
4
votes
2answers
263 views

Find this limit without using L'Hospital's rule

I have to find this limit without using l'Hôspital's rule: $$\lim_{x\to 0} \frac{\alpha \sin \beta x - \beta \sin \alpha x}{x^2 \sin \alpha x}$$ Using L'Hôspital's rule gives: ...
4
votes
3answers
2k views

How do we find specific values of sin and cos given the series definition

$\exp:x\mapsto \sum\limits_{n=0}^{+\infty}\cfrac{1}{n!}x^n$ $\cos:x\mapsto \Re\left(\exp \left(i x\right)\right)=\sum\limits_{n=0}^{+\infty}\cfrac{\left(-1\right)^n}{\left(2n\right)!}x^{2n}$ ...
4
votes
7answers
8k views

Proof of $\arctan(x) = \arcsin(x/\sqrt{1+x^2})$

I've tried following this way, but I haven't succeeded. Thank you!
4
votes
3answers
350 views

Solving $E=\frac{1}{\sin10^\circ}-\frac{\sqrt3}{\cos10^\circ}$

$$E=\frac{1}{\sin10^\circ}-\frac{\sqrt3}{\cos10^\circ}$$ I got no idea how to find the solution to this. Can someone put me on the right track? Thank you very much.
4
votes
4answers
403 views

A trigonometric identity: $(\sin x)^{-2}+(\cos x)^{-2}=(\tan x+\cot x)^2$

I've been trying to prove it for a while, but can't seem to get anywhere. $$\frac{1}{\sin^2\theta} + \frac{1}{\cos^2\theta} = (\tan \theta + \cot \theta)^2$$ Could someone please provide a valid ...
3
votes
2answers
603 views

Proving: $\cos A \cdot \cos 2A \cdot \cos 2^{2}A \cdot \cos 2^{3}A … \cos 2^{n-1}A = \frac { \sin 2^n A}{ 2^n \sin A } $

$$\cos A \cdot \cos 2A \cdot \cos 2^{2}A \cdot \cos 2^{3}A ... \cos 2^{n-1}A = \frac { \sin 2^n A}{ 2^n \sin A } $$ I am very much inquisitive to see how this trigonometrical identity can be ...
2
votes
3answers
107 views

Convergence test of the series $\sum\sin100n$ [duplicate]

I need to prove that $$\sum_{n=1}^{\infty} {\sin{100n}} \; \text{diverges}$$ I think the best way to do it is to show that $\lim_{n\to \infty}{\sin{100n}}\not=0$. But how do I prove it?
2
votes
3answers
1k views

How can I find the following product? $ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$

How can I find the following product using elementary trigonometry? $$ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$$ I have tried using a substitution, but nothing has worked.
2
votes
2answers
488 views

Mean Value Theorem: $\frac{2}{\pi}<\frac{\sin x}{x}<1$

I need to show that $\dfrac{2}{\pi}<\dfrac{\sin(x)}{x}<1$ for $0<x<\dfrac{\pi}{2}$. I know I need to use the mean value theorem, would I just say that since $f$ is continuous in the ...
2
votes
3answers
686 views

If $\sin(a)\sin(b)\sin(c)+\cos(a)\cos(b)=1$ then find the value of $\sin(c)$

If $$\sin(a)\sin(b)\sin(c)+\cos(a)\cos(b)=1,$$;where abc are the angles of the triangle.! then find the value of $\sin(c)$. By trial and error put this triangle as right angled isosceles and got the ...
2
votes
4answers
2k views

How to derive inverse hyperbolic trigonometric functions

$e^{i\theta}=\cos\theta + i\sin \theta$ $e^{i\sin^{-1}x}=\cos(\sin^{-1}x)+i\sin(\sin^{-1}x)$ $i\sin^{-1}x=\ln|\sqrt{1-x^2} + ix|$ $\sin^{-1}x=-i\ln|\sqrt{1-x^2} + ix|$ Now from here I'm kind of ...
2
votes
0answers
366 views

Product of Sine: $\prod_{i=1}^n\sin x_i=k$

From the article Products of Sines, we have $\sin 15^\circ\sin75^\circ=\sin 18^\circ\sin54^\circ=\frac{1}{4}$. We can rewrite this as $\sin \frac{\pi}{12}\sin\frac{5\pi}{12}=\sin ...
1
vote
2answers
82 views

Problem when $x=\cos (a) +i\sin(a),\ y=\cos (b) +i\sin(b),\ z=\cos (c) +i\sin(c),\ x+y+z=0$

Problem: If $$x=\cos (a) +i\sin(a),\ y=\cos (b) +i\sin(b),\ z=\cos (c) +i\sin(c),\ x+y+z=0$$ then which of the following can be true: 1) $\cos 3a + \cos 3b + \cos 3c = 3 \cos (a+b+c)$ 2) $1+\cos ...
0
votes
2answers
307 views

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

And when does $\sin^{-1}(\sin(x)) = x$
0
votes
1answer
755 views

How to prove $ \sin x=…(1+\frac{x}{3\pi})(1+\frac{x}{2\pi})(1+\frac{x}{\pi})x(1-\frac{x}{3\pi})(1-\frac{x}{2\pi})(1-\frac{x}{\pi})…$? [duplicate]

Possible Duplicate: infinite product of sine function Here is an other one which is more or less what Euler did in one of his proofs. The function sinx where x∈R is zero exactly at ...
26
votes
5answers
4k views

Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small?

I haven't touched Physics and Math (especially continuous Math) for a long time, so please bear with me. In essence, I'm going over a few Physics lectures, one which tries to calculate the Force ...
12
votes
3answers
274 views

Existence of continuous angle function $\theta:S^1\to\mathbb{R}$

Let $S^1\subseteq\mathbb{C}$ be the unit circle and let $U\subseteq S^1$ be open. How to show that there exist a continuous function $$\theta:U\to\mathbb{R}$$ such that $$e^{i\theta(z)}=z$$ for all ...
12
votes
3answers
241 views

Could we show $1-(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dots)^2=(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}- \dots)^2$ if we didn't know about Taylor Expansion?

Suppose that humanity haven't discovered Taylor Series Expansion of trigonometric functions or of any function that would help us on this. Which means we are not allowed to replace the given infinite ...
10
votes
4answers
177 views

New very simple golden ratio construction incorporating a triangle, square, and pentagon all with sides of equal length. Is there any prior art?

Consider three regular polygons with 3, 4, and 5 sides wherein all the polygons have sides of equal length X throughout, as illustrated below. The ratio of the red line segment a to the blue line ...
10
votes
6answers
513 views

Derive $\frac{d}{dx} \left[\sin^{-1} x\right] = \frac{1}{\sqrt{1-x^2}}$

Derive $\frac{d}{dx} \left[\sin^{-1} x\right] = \frac{1}{\sqrt{1-x^2}}$ (Hint: set $x = \sin y$ and use implicit differentiation) So, I tried to use the hint and I got: $x = \sin y$ ...
8
votes
2answers
292 views

Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$

$$\int_0^1 \frac{\arcsin(x)}{x}dx$$ This is a proposed for a Calculus II exam, and I have absolutely no idea how to solve it. Tried using Frullani or Lobachevsky integrals, or beta and gamma ...
7
votes
4answers
344 views

Trigonometric Equation $\sin x=\tan\frac{\pi}{15}\tan\frac{4\pi}{15}\tan\frac{3\pi}{10}\tan\frac{6\pi}{15}$

How can I solve this trigonometric equation? $$\sin x=\tan\frac{\pi}{15}\tan\frac{4\pi}{15}\tan\frac{3\pi}{10}\tan\frac{6\pi}{15}$$
7
votes
2answers
7k views

Sum of $\cos(k x)$ [duplicate]

I'm trying to calculate the trigonometric sum : $$\sum\limits_{k=1}^{n}\cos(k x)$$ This is what I've tried so far : $$\renewcommand\Re{\operatorname{Re}} \begin{align*} \sum\limits_{k=1}^{n}\cos(k x) ...
6
votes
1answer
531 views

Rigorous proof of an infinite product.

I'll give a proof of the following expansion: $$\frac{\sin x}{x} = \prod_{i=1}^{\infty} \cos \frac{x}{2^i}$$ $${\sin x} = 2 \cos \frac{x}{2}\sin \frac{x}{2}$$ $${\sin x} = 2^2 \cos \frac{x}{2}\cos ...
5
votes
4answers
137 views

How to simpify $\cos x - \sin x$

How does one simplify $$\cos x - \sin x$$ I tried multiplying by $\cos x + \sin x$, but that just gets me $$\cos x - \sin x = \frac{\cos 2x}{\cos x + \sin x}$$ which is worse. Yet ...
3
votes
6answers
210 views

What is the value of $\frac{\sin x}x$ at $x=0$?

On plotting graph for $\frac{\sin x}{x}$ using Wolfram|Alpha and Google, got that : also, I can get the value of $\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$ using squeeze theorem and as illustrated ...
3
votes
3answers
756 views

Evaluate the limit of $\ln(\cos 2x)/\ln (\cos 3x)$ as $x\to 0$

Evaluate Limits $$\lim_{x\to 0}\frac{\ln(\cos(2x))}{\ln(\cos(3x))}$$ Method 1 :Using L'Hopital's Rule to Evaluate Limits (indicated by $\stackrel{LHR}{=}$. LHR stands for L'Hôpital Rule) ...
3
votes
3answers
179 views

Integrating trigonometric function problem $\int \frac{3\sin x+2\cos x}{2\sin x+3\cos x}dx$ [duplicate]

\begin{eqnarray*} \int \frac{3\sin x+2\cos x}{2\sin x+3\cos x}dx &=& \int \frac{(3\sin x+2\cos x)/\cos x}{(2\sin x+3\cos x)/\cos x}dx\\ \\ &=& \int \frac{3\tan x +2}{2\tan x +3} dx\\ ...