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

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7
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2answers
1k 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 ...
7
votes
2answers
153 views

Show that $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$

Show that $$\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$$
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} \frac{1}...
5
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
353 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
3answers
2k views

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

$\exp:x\mapsto \sum\limits_{n=0}^{+\infty}\cfrac{1}{n!}x^n$ $\cos:x\mapsto \Re\left(\exp \left(i x\right)\right)=\sum\limits_{n=0}^{+\infty}\cfrac{\left(-1\right)^n}{\left(2n\right)!}x^{2n}$ $\sin:x\...
4
votes
4answers
407 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
280 views

Concerning the sequence $\Big(\dfrac {\tan n}{n}\Big) $

Is the sequence $\Big(\dfrac {\tan n}{n}\Big) $ convergent ? If not convergent , is it properly divergent i.e. tends to either $+\infty$ or $-\infty$ ? ( Owing to $\tan (n+1)= \dfrac {\tan n + \tan ...
3
votes
2answers
607 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
2answers
490 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
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.
0
votes
2answers
314 views

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

And when does $\sin^{-1}(\sin(x)) = x$
0
votes
1answer
760 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 x=...
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 ...
12
votes
3answers
283 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 $z\...
10
votes
4answers
223 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
515 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: $x = \sin y$ $\frac{d}{dx}\...
10
votes
5answers
10k views

Calculate the area of the crescent

I found this problem on a thread on Stack overflow where it was posted as "job interview question". Unfortunately I cannot find the question. But I saved the picture and just cannot figure it out. ...
8
votes
2answers
293 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
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) ...
7
votes
4answers
345 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? $$\sin x=\tan\frac{\pi}{15}\tan\frac{4\pi}{15}\tan\frac{3\pi}{10}\tan\frac{6\pi}{15}$$
6
votes
2answers
295 views

Proof that $\lim_{n\to\infty}{\sin{100n}}$ does not exist

How to prove that $$\lim_{n\to\infty}{\sin{100n}}$$ doesn't exist? Some possible approaches: It would be enough to find two subsequences $n_{k}$ that converge to two different numbers. But it's ...
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 ...
4
votes
4answers
873 views

What is the most elegant and simple proof for the law of cosines?

Given 2 sides, and an angle between those two sides, what is the simplest proof you can come up with to find the measure of the 3rd side?
3
votes
4answers
516 views

Given that $\;\sin^3x\sin3x = \sum^n_{m=0}C_m\cos mx\,,\; C_n \neq 0\;$ is an identity . Find the value of n.

Problem : Given that $\sin^3 x \sin 3x = \sum^n_{m=0}C_m \cos mx, C_n \neq 0 $ is an identity. Find the value of n. I tried : $\sin3x = 3\sin x - 4\sin^3 x$ but unable to reach to any point.... ...
3
votes
3answers
808 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
155 views

Prove an inequality with a $\sin$ function: $\sin(x) > \frac2\pi x$ for $0<x<\frac\pi2$

$$\forall{x\in(0,\frac{\pi}{2})}\ \sin(x) > \frac{2}{\pi}x $$ I suppose that solving $ \sin x = \frac{2}{\pi}x $ is the top difficulty of this exercise, but I don't know how to think out such ...
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
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\\ &...
2
votes
2answers
92 views

Relationship among $A,B,C,D$ for $\cos A\cos B=\cos C\cos D$

While solving this Question, I could derive the following: As $\displaystyle 2\cos A\cos B=\cos(A-B)+\cos(A+B)$ substituting $A+B=90^\circ\iff B=90^\circ-A$ we get $\displaystyle 2\cos A\cos(90^\...
1
vote
2answers
53 views

How to to a better approach for this :?

If, $$x\cos A+y\sin A=k=x\cos B+y\sin B$$ Then find $(\cos A)(\cos B)$, $(\sin A)(\sin B)$ and $\cos A+\cos B$ and express them in terms of $x,y,k$ I found a solution but it included a really ...
1
vote
4answers
129 views

How to Find the Minimum Value for $\displaystyle\frac{\tan x}{x}$ (piecewise defined)?

$$F (x) = \begin{cases} \displaystyle \frac{\tan(x)}{x} & x \not=0 \\ 1 & x=0 \end{cases} $$ How do I prove that there is a local minimum at $x=0$?
1
vote
1answer
139 views

Help evaluating $\int_{-1/2}^{1/2}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)$

$$\int_{-1/2}^{1/2}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)df$$ I've tried simplifying the integrand, but I can't get to a point where I can evaluate the integral. I know $\prod_{i=0}^...
7
votes
3answers
514 views

Prove trigonometry identity for $\cos A+\cos B+\cos C$

I humbly ask for help in the following problem. If \begin{equation} A+B+C=180 \end{equation} Then prove \begin{equation} \cos A+\cos B+\cos C=1+4\sin(A/2)\sin(B/2)\sin(C/2) \end{equation} How would ...
7
votes
3answers
5k views

Prove that the Tangent of 75 degrees equals 2 plus the square-root of 3

My (very simple) question to a friend was how do I prove the following using basic trig principles: $\tan75^\circ = 2 + \sqrt{3}$ He gave this proof (via a text message!) $1. \tan75^\circ$ $2. = \...
6
votes
3answers
1k views

Ambiguity of notation: $\sin(x)^2$

Several people have told me that $\sin(x)^2 = \sin(x^2)$. However, on several computing platforms, such as the TI-84 and Wolfram|Alpha, $\sin(x)^2 = \sin^2(x)$. Can I safely conclude that the notation ...
5
votes
2answers
370 views

Trig integral $\int{ \cos{x} + \sin{x}\cos{x} dx }$

Assume we have: $ \int{ \cos{x} + \sin{x}\cos{x} dx } $ 2 ways to do it: Use $\sin{x}\cos{x} = \frac{ \sin{2x} }{2} $ Then $ \int{ \cos{x} + \frac{\sin{2x}}{2} dx } $ $ = \sin{x} - \frac{ cos{2x} ...
5
votes
2answers
2k views

Identity for a weighted sum of sines / sines with different amplitudes

I'm trying to simplify the following sum of sines with different amplitudes $$ a \sin(\theta) + b \sin(\phi) = ??? \,\,\,\,\, (1) $$ I know that $$ a \sin(\theta) + a \sin(\phi) = a\cdot2\sin\left(\...
4
votes
3answers
274 views

Differentiate $\sin \sqrt{x^2+1} $with respect to $x$?

Differentiate $$ \sin \sqrt{x^2+1} $$ with respect to $x$? Can someone please help me with question, im very lost.
4
votes
5answers
121 views

Solving $\sin\theta -1=\cos\theta $

Solve$$\sin\theta -1=\cos\theta $$ Steps I took to solve this: $$\sin^{ 2 }\theta -2\sin\theta +1=1-\sin^2\theta $$ $$2\sin^{ 2 }\theta -2\sin\theta =0$$ $$(2\sin\theta )(\sin\theta -1)=0$$ $$2\...
3
votes
1answer
109 views

For which integer $n$, $\sin\left(\frac{\pi}{n}\right)$ can be a rational?

When I studying the trigonometric functions, I sow that most of the values of $\sin\left(\dfrac{\pi}{n}\right)$ and $\cos\left(\dfrac{\pi}{n}\right)$ where $n\in\mathbb{N}$ are irrational. How can we ...
3
votes
2answers
337 views

Prove that $x^2<\sin x \tan x$ as $x \to 0$ [duplicate]

$$x^2<\sin x \tan x \quad as \; x \to 0$$ I made the substitution $x \to \arctan x$ . $\arctan^2 x<x\sin (\arctan x)$ $\arctan x < \large \frac{x}{(x^2+1)^{\frac 14}}$ There are two ...
2
votes
2answers
90 views

Find the product

Task is to find $$\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}).$$ I tried to make it with double-angle formula: $\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}) = \frac{\prod_{k=0}^\infty \sin(x2^{1-k})}{2^\...
2
votes
4answers
230 views

Number of iterations to reach cosine's fixed point

I was messing around with my calculator the other day when I saw something interesting happen. Whenever I repetitively took the cosine of any number, it always ended up on a particular number (Dottie'...
1
vote
1answer
60 views

Simple Golden Ratio Construction with Three Lines, and Interesting Implied Circle?

Consider three equal lines (as illustrated below). A red, green, and orange line of equal length all rest upon the same horizontal line. The red line is stood upon its end in a manner perpendicular ...
0
votes
2answers
139 views

How is $ \cos (\alpha / \beta) $ expressed in terms of $\cos \alpha $ and $ \cos \beta $?

If it is possible to express $ \cos n \alpha $ in terms of $ \cos \alpha $ as a power series for integer $n$ ... I like to see an expression for the quotient angle that obviously tallies when $ (\...
0
votes
1answer
104 views

Engineering Mathematics

I have been revising basic compound angles and I am struggling to understand the following question from the examples I have previously studied on such topic. I cannot envisage the compound angle ...
-1
votes
5answers
120 views

If $\cos \theta = \cos \alpha$, what can we conclude about $\theta - \alpha$?

Suppose $\cos \theta = \cos \alpha$. If we give no restrictions to the angles, I understand that we must $\mathbf{not}$ conclude that $\theta = \alpha $. Can we conclude then that $\theta - \alpha = 2 ...