Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.
52
votes
12answers
5k 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 ...
9
votes
2answers
2k 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:
$$\large \sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n ...
20
votes
2answers
1k views
Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $
Is there any way to show that
$$\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( ...
23
votes
15answers
5k 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
1k 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:
...
2
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 ...
7
votes
4answers
2k 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 ...
41
votes
6answers
2k 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 ...
27
votes
7answers
9k 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 ...
15
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
$$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
16
votes
1answer
1k views
Infinite product of sine function
How to prove the following product?
$$\frac{\sin(x)}{x}=
\left(1+\frac{x}{\pi}\right)
\left(1-\frac{x}{\pi}\right)
\left(1+\frac{x}{2\pi}\right)
\left(1-\frac{x}{2\pi}\right)
...
9
votes
1answer
633 views
$\arcsin$ written as $\sin^{-1}(x)$
I know that different people follow different conventions, but whenever I see $\arcsin(x)$ written as $\sin^{-1}(x)$, I find myself thinking it wrong, since $\sin^{-1}(x)$ should be $\csc(x)$, and not ...
51
votes
6answers
4k views
Ways to evaluate $\int \sec \theta \, d \theta$
The standard approach for showing $\int \sec \theta \, d \theta = \ln |\sec \theta + \tan \theta| + C$ is to multiply by $\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then do a ...
22
votes
5answers
1k views
Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?
I have seen the Fresnel integral
$$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$
evaluated by contour integration and other complex analysis methods, and I have found these methods to be the ...
19
votes
4answers
1k views
Prove that $\sum_{k=1}^{n-1}\tan^{2}\frac{k \pi}{2n} = \frac{(n-1)(2n-1)}{3}$
How can you prove that:
$$\sum_{k=1}^{n-1}\tan^{2}\frac{k \pi}{2n} = \frac{(n-1)(2n-1)}{3}$$
for every integer $n\geq 1$.
PS: no, it's not a homework... :-)
16
votes
3answers
488 views
Integrate $\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$
Evaluate the integral $$\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 ...
6
votes
3answers
2k 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
888 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
635 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)}$?
9
votes
5answers
360 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}\\
\\
...
9
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 ...
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
2answers
567 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)$$
however, only for $uv<1$. My question is: where does this condition come from? The ...
4
votes
3answers
627 views
How quickly we forget - basic trig. Calculate the area of a polygon
I think the easiest way to do this is with trigonometry, but I've forgotten most of the maths I learnt in school. I'm writing a program (for demonstrative purposes) that defines a Shape, and ...
36
votes
4answers
746 views
Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$
Prove that $$\sum _{l=1}^{n}\sum _{k=1}^{n-1}\tan \frac {lk\pi } {2n+1}\tan \frac {l( k+1) \pi } {2n+1}=0$$
It is very easy to prove this identity for each fixed $n$ . For example let $n = 6$; ...
20
votes
7answers
3k views
How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?
I would like to find the apothem of a regular pentagon. It follows from
$$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$
But how can this be proved (geometrically or trigonometrically)?
5
votes
6answers
2k views
Need help in proving that $\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$
We need to prove that $$\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$$
I have tried and it gets confusing.
0
votes
1answer
623 views
$\sum \cos$ when angles are in arithmetic progression [duplicate]
Possible Duplicate:
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?
Prove $$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + ...
20
votes
6answers
2k views
How to prove that: $\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}$
How can we prove the following trigonometric identity?
$$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$
37
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?
$$\displaystyle\int_{0}^{\frac{\pi }{2}}{{{x}^{2}}\sqrt{\tan x}\sin \left( 2x \right)\text{d}x}$$
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 ...
4
votes
1answer
361 views
rational angles with sines expressible with radicals
An angle x is rational when measured in degrees. sin(x) is can be written using radicals. What are the conditions on x? If nested square roots are allowed?
What I know so far:
If sin(x) can be ...
10
votes
2answers
1k views
When is $\sin x$ an algebraic number and when is it non-algebraic?
Show that if $x$ is rational, then $\sin x$ is algebraic number when $x$ is in degrees and $\sin x$ is non algebraic when $x$ is in radians.
Details: so we have $\sin(p/q)$ is algebraic when ...
10
votes
1answer
474 views
Proving that $ \frac{1}{\sin(45°)\sin(46°)}+\frac{1}{\sin(47°)\sin(48°)}+…+\frac{1}{\sin(133°)\sin(134°)}=\frac{1}{\sin(1°)}$
I would like to show that the following trigonometric sum
$$ \frac{1}{\sin(45°)\sin(46°)}+\frac{1}{\sin(47°)\sin(48°)}+\cdots+\frac{1}{\sin(133°)\sin(134°)}$$
...
10
votes
2answers
317 views
Inequality $\sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0$
Show the following inequality for any $x\in [0, \pi]$ and $n\in \mathbb{N}^*$,
$$
\sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0.
$$
I have this question a very long time ago from a book or magazine but I ...
5
votes
6answers
316 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 ...
5
votes
1answer
720 views
Sine values being rational
Can $$\sin r\pi $$ be rational if $r$ is irrational? Either a direct or existence proof is fine.
11
votes
3answers
2k views
How to prove Lagrange trigonometric identity [duplicate]
I would to prove that
$$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+
\frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$
given that
...
7
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 $\sin$:
$$\int e^{\alpha x}\sin(\beta x) ...
5
votes
2answers
740 views
Power-reduction formula
According to the Power-reduction formula, one can interchange between $\cos(x)^n$ and $\cos(nx)$ like the following:
$$
\cos^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} \binom{n}{k} ...
2
votes
4answers
189 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 - ...
2
votes
5answers
374 views
Why $x<\tan{x}$ while $0<x<\frac{\pi}{2}$?
In proof of $\displaystyle\lim_{x\rightarrow0}\frac{\sin{x}}{x}=1$ is assumed that $\sin{x}\leq{x}\leq\tan{x}$ while $0<x<\frac{\pi}{2}$. First comparison is clear, arc length must be greater ...
4
votes
3answers
390 views
How to derive compositions of trigonometric and inverse trigonometric functions?
To prove:
$$\sin({\arccos{x}})=\sqrt{1-x^2}$$
$$\cos{\arcsin{x}}=\sqrt{1-x^2}$$
$$\sin{\arctan{x}}=\frac{x}{\sqrt{1+x^2}}$$
$$\cos{\arctan{x}}=\frac{1}{\sqrt{1+x^2}}$$
...
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
$$ \prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\pi n/2)}{2^{n-1}}$$
14
votes
5answers
920 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 ...
7
votes
5answers
5k 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 ...
13
votes
2answers
327 views
How to do a very long division: continued fraction for tan
I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
9
votes
11answers
3k 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 ...
