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

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125
votes
14answers
13k 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 ...
24
votes
3answers
4k 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 ...
4
votes
1answer
1k 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 + ...
26
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( ...
22
votes
3answers
4k views

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

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
34
votes
2answers
3k views

What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?

I tried and got this $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an ...
37
votes
7answers
14k 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 ...
29
votes
15answers
7k 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?) ...
4
votes
6answers
5k 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 ...
20
votes
3answers
823 views

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

Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$
33
votes
5answers
2k 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 ...
63
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 ...
5
votes
3answers
3k 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: ...
10
votes
8answers
2k 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 ...
23
votes
4answers
2k 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... :-)
7
votes
3answers
756 views

How to derive compositions of trigonometric and inverse trigonometric functions?

To prove: $$\begin{align} \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}}\\ ...
6
votes
4answers
359 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
7
votes
3answers
4k 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 ...
5
votes
8answers
2k views

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

I want to prove \begin{equation*} \tan A + \tan B + \tan C = \tan A\tan B\tan C,$ $A+B+C = 180^\circ \end{equation*} We know that \begin{equation*} \tan(A+B) = \frac{\tan A+\tan B}{1-\tan A\tan ...
49
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 ...
18
votes
1answer
3k 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) ...
16
votes
4answers
2k 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 $$ \prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\pi n/2)}{2^{n-1}}$$
14
votes
1answer
877 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 ...
12
votes
3answers
377 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 ...
3
votes
5answers
2k views

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

$$\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$$ I have attempted this question by expanding the left side using the cosine sum and difference formulas and then multiplying, and then ...
7
votes
3answers
3k views

Solving trigonometric equations of the form $a\sin x + b\cos x = c$

Suppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 < x < 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos ...
8
votes
3answers
841 views

Finding the limit $\lim \limits_{n \to \infty}\ (\cos \frac x 2 \cdot\cos \frac x 4\cdot \cos \frac x 8\cdots \cos \frac x {2^n}) $

This limit seemed quite unusual to me as there aren't any intermediate forms or series expansions which are generally used in limits. Stuck on this for a while now .Here's how it goes : $$ \lim ...
8
votes
3answers
763 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 $\displaystyle {(a + bi)}^{(c + di)}$?
6
votes
6answers
4k 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.
2
votes
4answers
259 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 - ...
39
votes
4answers
897 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 $n = 6$; ...
21
votes
8answers
4k 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)?
11
votes
5answers
438 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}\\ \\ ...
4
votes
2answers
2k 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 ...
2
votes
2answers
1k views

Proof for the formula of sum of arcsine functions $ \arcsin x + \arcsin y $

It is known that the following holds good: $$ \arcsin x + \arcsin y \\ \begin{align} &=\arcsin( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \;\;;x^2+y^2 \le 1 \;\text{ or }\; x^2+y^2 > 1, xy< 0\\ ...
7
votes
2answers
5k 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 ...
1
vote
3answers
382 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 : ...
24
votes
6answers
4k views

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

How can we prove the following trigonometric identity? $$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$
22
votes
9answers
8k views

Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
15
votes
5answers
3k 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 ...
8
votes
5answers
7k views

how to strictly prove $\sin x<x$ for $0<x<\frac{\pi}{2}$

$$\sin x<x\,(0<x<\frac{\pi}{2})$$ In most textbooks, to prove this inequality is based on geometry illustration (draw a circle, compare arc length and chord ), but I think that strict proof ...
16
votes
5answers
10k 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: $$ ...
3
votes
4answers
873 views

Solving $2x - \sin 2x = \pi/2$ for $0 < x < \pi/2$

What is $x$ in closed form if $2x-\sin2x=\pi/2$, $x$ in the first quadrant?
102
votes
3answers
6k views

A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I

The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I ...
55
votes
6answers
5k views

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

I'm having trouble computing the integral: $$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$$ I hope that it can be expressed in terms of elementary functions. I've tried simple substitutions such as ...
17
votes
11answers
5k 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 ...
13
votes
3answers
1k views

When the trig functions moved from the right triangle to the unit circle?

I have to write a paper about the unit circle and I'm trying to uncover some of its origins. Also, when the trig functions were expanded to angles greater than 90° and what was the rationale behind ...
15
votes
5answers
3k 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 ...
12
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 ...