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

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82
votes
13answers
8k views

How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim\limits_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my ...
14
votes
2answers
3k views

How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?

How can we sum up $\sin$ and $\cos$ series when the angles are in A.P (arithmetic progression) ?For example here is the sum of $\cos$ series: $$\large \sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n ...
23
votes
2answers
2k views

Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $

Is there any way to show that $$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^k}\left( ...
3
votes
1answer
954 views

$\sum \cos$ when angles are in arithmetic progression [duplicate]

Possible Duplicate: How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? Prove $$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + ...
3
votes
6answers
3k views

Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $

How would I verify the following double angle identity. $$ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $$ So far I have done this. $$ (\sin A\cos B+\cos A\sin B)(\sin A\cos B-\cos A\sin B) $$But I am not sure ...
34
votes
7answers
11k views

How can I understand and prove the “sum and difference formulas” in trigonometry? (cos(a ± b) = …, etc.)?

The "sum and difference" formulas often come in handy, but it's not immediately obvious that they would be true. \begin{align} \sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha ...
20
votes
2answers
3k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using n_th root of unity $$\Large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
60
votes
8answers
5k views

Ways to evaluate $\int \sec \theta \, \mathrm d \theta$

The standard approach for showing $\int \sec \theta \, \mathrm d \theta = \ln |\sec \theta + \tan \theta| + C$ is to multiply by $\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then ...
7
votes
4answers
3k views

When is $\sin(x)$ rational?

Obviously, there are some points (like $\pi,30$) but I am unsure if there are more. How can it be proved that there are no more points, or what those points will be? EDIT: I largely meant to ask ...
27
votes
15answers
6k views

Intuitive understanding of the derivatives of $\sin x$ and $\cos x$

One of the first things ever taught in a differential calculus class: The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$. This leads to a rather neat (and convenient?) ...
5
votes
3answers
2k views

Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
46
votes
6answers
3k views

How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?

How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$ I found the above interesting identity in the book $\bf \pi$ Unleashed. Does anyone knows how to ...
22
votes
5answers
1k views

Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?

I have seen the Fresnel integral $$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$ evaluated by contour integration and other complex analysis methods, and I have found these methods to be the ...
16
votes
1answer
2k views

Infinite product of sine function

How to prove the following product? $$\frac{\sin(x)}{x}= \left(1+\frac{x}{\pi}\right) \left(1-\frac{x}{\pi}\right) \left(1+\frac{x}{2\pi}\right) \left(1-\frac{x}{2\pi}\right) ...
11
votes
1answer
758 views

$\arcsin$ written as $\sin^{-1}(x)$

I know that different people follow different conventions, but whenever I see $\arcsin(x)$ written as $\sin^{-1}(x)$, I find myself thinking it wrong, since $\sin^{-1}(x)$ should be $\csc(x)$, and not ...
21
votes
4answers
1k views

Prove that $\sum_{k=1}^{n-1}\tan^{2}\frac{k \pi}{2n} = \frac{(n-1)(2n-1)}{3}$

How can you prove that: $$\sum_{k=1}^{n-1}\tan^{2}\frac{k \pi}{2n} = \frac{(n-1)(2n-1)}{3}$$ for every integer $n\geq 1$. PS: no, it's not a homework... :-)
17
votes
3answers
574 views

Integrate $\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$

Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$
7
votes
7answers
1k views

Different definitions of trigonometric functions

In school, we learn that sin is "opposite over hypotenuse" and cos is "adjacent over hypotenuse". Later on, we learn the power series definitions of sin and cos. How can one prove that these two ...
11
votes
3answers
307 views

Double limit of $\cos^{2n}(m! \pi x)$ at rationals and irrationals

I stumbled upon this "relation" (is the name correct?): $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&x\text{ is rational}\\ 0,&x\text{ is ...
7
votes
3answers
3k views

Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle

Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle This question came up in a miscellaneous problem set I have been working on to refresh my ...
3
votes
7answers
1k views

Prove that $\tan A + \tan B + \tan C = \tan A\tan B\tan C,$ $A+B+C = 180^\circ$

RTP: $\tan A + \tan B + \tan C = \tan A\tan B\tan C,$ $A+B+C = 180^\circ$ Well, we know that $\tan(A+B) = \frac{\tan A+\tan B}{1-\tan A\tan B}$ and that $A+B = 180^\circ-C.$ Therefore $\tan(A+B) ...
8
votes
3answers
705 views

How to raise a complex number to the power of another complex number?

How do I calculate the outcome of taking one complex number to the power of another, ie $\displaystyle {(a + bi)}^{(c + di)}$?
20
votes
7answers
3k views

How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?

I would like to find the apothem of a regular pentagon. It follows from $$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$ But how can this be proved (geometrically or trigonometrically)?
2
votes
4answers
233 views

Show that $2\tan^{-1}(2) = \pi - \cos^{-1}(\frac{3}{5})$

Show that $2\tan^{-1}(2) = \pi - \cos^{-1}(\frac{3}{5})$ So, taking $\tan$ of both sides I get: LHS $=\frac{2\tan(\tan^{-1}(2))}{1 - \tan^2(\tan^{-1}(2))} = -\frac{4}{3}$ and RHS $= \tan(\pi - ...
11
votes
5answers
412 views

Prove the trigonometric identity $(35)$

Prove that \begin{equation} \prod_{k=1}^{\lfloor (n-1)/2 \rfloor}\tan \left(\frac{k \pi}{n}\right)= \left\{ \begin{aligned} \sqrt{n} \space \space \text{for $n$ odd}\\ \\ ...
10
votes
5answers
2k views

Understanding imaginary exponents

Greetings! I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean? I've read a few pages on this issue, and they all seem to ...
4
votes
2answers
873 views

A question about the arctangent addition formula.

In the arctangent formula, we have that: $$\arctan{u}+\arctan{v}=\arctan\left(\frac{u+v}{1-uv}\right)$$ however, only for $uv<1$. My question is: where does this condition come from? The ...
6
votes
6answers
3k views

Need help in proving that $\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$

We need to prove that $$\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$$ I have tried and it gets confusing.
16
votes
4answers
1k views

How to prove those “curious identities”?

How to prove $$ \prod_{k=1}^{n-1} \sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}}$$ and $$ \prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\pi n/2)}{2^{n-1}}$$
15
votes
11answers
4k views

Prove $\sin^2\theta + \cos^2\theta = 1$

How do you prove the following trigonometric identity: $$ \sin^2\theta+\cos^2\theta=1$$ I'm curious to know of the different ways of proving this depending on different characterizations of sine and ...
14
votes
5answers
2k views

$\sin(A)$, where $A$ is a matrix

If $A$ is an $n\times n$ matrix with elements $a_{ij}$ $i=$i'th row, $j=$j'th column. Then $e^A$ is also a matrix as can be seen by expanding it in a power series.Is $e^A$ always convergent and ...
5
votes
4answers
262 views

showing $\arctan(\frac{2}{3}) = \frac{1}{2} \arctan(\frac{12}{5})$

I'm having problem with showing that: $$\arctan(\frac{2}{3}) = \frac{1}{2} \arctan(\frac{12}{5})$$ I would need some help in the right direction
10
votes
7answers
1k views

Solve trigonometric equation: $1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$

Dealing with a physics Problem I get the following equation to solve for $\alpha$ $1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$ Putting this in Mathematica gives the result: $a==2 ...
4
votes
3answers
512 views

How to derive compositions of trigonometric and inverse trigonometric functions?

To prove: $$\sin({\arccos{x}})=\sqrt{1-x^2}$$ $$\cos{\arcsin{x}}=\sqrt{1-x^2}$$ $$\sin{\arctan{x}}=\frac{x}{\sqrt{1+x^2}}$$ $$\cos{\arctan{x}}=\frac{1}{\sqrt{1+x^2}}$$ ...
4
votes
3answers
724 views

How quickly we forget - basic trig. Calculate the area of a polygon

I think the easiest way to do this is with trigonometry, but I've forgotten most of the maths I learnt in school. I'm writing a program (for demonstrative purposes) that defines a Shape, and ...
1
vote
3answers
288 views

Simplifying an Arctan equation

Simplify $f(x)=Arctan(x)+Arctan(\frac{1}{x})$. My attempt For all non null real numbers $f$ is derivable and is odd: so for all non null real number x : ...
48
votes
5answers
2k views

Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$

I've got troubles in computing the below integral: $$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$$ I hope it can be expressed in elementary functions. I've tried simple substitution as $u=\sin(x)$ ...
37
votes
4answers
783 views

Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$

Prove that $$\sum _{l=1}^{n}\sum _{k=1}^{n-1}\tan \frac {lk\pi } {2n+1}\tan \frac {l( k+1) \pi } {2n+1}=0$$ It is very easy to prove this identity for each fixed $n$ . For example let $n = 6$; ...
15
votes
7answers
1k views

Proving $\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$

How do I show that: $$\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$$ This is actually problem B $4371$ given at this link. Looks like ...
8
votes
5answers
7k views

How to expand $\cos nx$ with $\cos x$?

Multiple Angle Identities: How to expand $\cos nx$ with $\cos x$, such as $$\cos10x=512(\cos x)^{10}-1280(\cos x)^8+1120(\cos x)^6-400(\cos x)^4+50(\cos x)^2-1$$ See a list of trigonometric ...
8
votes
3answers
1k views

Name of this identity? $\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha^2+\beta^2}$

Again: $$\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha^2+\beta^2}$$ Also the one for $\sin$: $$\int e^{\alpha x}\sin(\beta x) ...
6
votes
7answers
584 views

Fundamental Theorem of Trigonometry

This is a pretty open ended question and I apologize, in advance, if this is not the place for it. But what do you recommend should be given the title of the Fundamental Theorem of Trigonometry and ...
13
votes
4answers
8k views

What is the length of a sine wave from $0$ to $2\pi$?

What is the length of a sine wave from $0$ to $2\pi$? Physically I would plot $$y=\sin(x),\quad 0\le x\le {2\pi}$$ and measure line length. I think part of the answer is to integrate this: $$ ...
2
votes
5answers
271 views

Prove that $\sin^2(A) - \sin^2(B) = \sin(A + B)\sin(A -B)$

This is my attempt: $$ \begin{align} & \phantom{={}}\sin^2(A) - \sin^2(B) = \sin(A + B)\sin(A -B) \\[8pt] & = (\sin(A)\cos(B)+\cos(A)\sin(B))(\sin(A)\cos(B) - \cos(A)\sin(B)) \\[8pt] & = ...
23
votes
6answers
3k views

How to prove that: $\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}$

How can we prove the following trigonometric identity? $$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$
38
votes
6answers
3k views

$\sin 1^\circ$ is irrational but how do I prove it in a slick way? And $\tan(1^\circ)$ is …

In the book 101 problems in Trigonometry, Prof. Titu Andreescu and Prof. Feng asks for the proof the fact that $\cos 1^\circ$ is irrational and he proves it. The proof proceeds by contradiction and ...
18
votes
2answers
2k views

A hard definite integral with trigonometric functions

How could we get a closed form for this one? $$\displaystyle\int_{0}^{\frac{\pi }{2}}{{{x}^{2}}\sqrt{\tan x}\sin \left( 2x \right)\text{d}x}$$
16
votes
3answers
883 views

A “fast” way for computing $ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $?

Which is the fastest paper-pencil approach to compute the product $$ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $$
11
votes
3answers
817 views

Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $

It's my first post here and I was wondering if someone could help me with evaluating the definite integral $$ \int_0^{\Large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $$ Thanks in ...
6
votes
5answers
421 views

Solve problem of trigonometry.

Prove that $\sin\frac{\pi}{14}$ is a root of $8x^3 - 4x^2 - 4x + 1=0$. I have no clue how to proceed and tried to prove that the whole equation becomes $0$ when $\sin\frac{\pi}{14}$ is placed in ...