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

learn more… | top users | synonyms (1)

104
votes
9answers
10k views

Why is cos(90)=0.4 in WebGL?

I'm a graphical artist who is completely out of my depth on this site. However, I'm dabbling in WebGL (3D software for internet browsers) and trying to animate a bouncing ball. Apparently we can ...
83
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 ...
83
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 ...
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 ...
54
votes
2answers
1k views

Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}\mathrm dx$

Consider the following integral: $$\mathcal{I}=\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}\mathrm dx,$$ where ...
50
votes
1answer
3k views

Trigonometric sums related to the Verlinde formula

Original question (see also the revised, possibly simpler, version below): Let $g > 1, r > 1$ be integers. Playing around with the Verlinde formula (see below), I came across the expression ...
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 $u=\sin(x)$ ...
46
votes
5answers
3k views

Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$?

Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this ...
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 ...
40
votes
4answers
2k views

Do “Parabolic Trigonometric Functions” exist?

The parametric equation $$\begin{align*} x(t) &= \cos t\\ y(t) &= \sin t \end{align*}$$ traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly, $$\begin{align*} x(t) ...
40
votes
3answers
906 views

Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$

Prove that $$\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$$
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 ...
38
votes
2answers
1k views

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
37
votes
4answers
785 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 $n = 6$; ...
37
votes
1answer
791 views

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

Why is the following function $f(n)$ constant for any natural number $n$? $$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt ...
35
votes
4answers
2k views

Integral $\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}\mathrm dx$$
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 ...
33
votes
2answers
2k views

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$$ It also can be represented as $$I=\int_0^{\pi/4}\frac{\phi^2}{\cos \phi\,\sqrt{\cos ...
33
votes
3answers
1k views

Making trigonometric substitutions rigorous

I've been tutoring some basic calculus, and it made me think about something pretty basic. Let me explain the problem by example: Say we are given the integral $\int \frac{x^2}{\sqrt{1-x^2}}\ ...
33
votes
2answers
1k views

Is there an interpretation for this trigonometric identity?

A while ago I came across the following identity in an online math forum (of which I don't remember the name): $$\tan\left(\frac{\pi}{11}\right)+4\sin\left(\frac{3\pi}{11}\right)=\sqrt{11}.$$ It is ...
32
votes
3answers
613 views

$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$

Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction. $$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
31
votes
2answers
1k views

Integral $\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)\mathrm dx$$
31
votes
2answers
688 views

Finding integer solutions for trigonometric equation $8\sin^2\left(\frac{(k+1)\pi}{n}\right)=n\sin\left(\frac{2\pi}{n}\right)$

I thought up the problem of finding a regular $n$-sided polygon that has a diagonal with lenght $d_k$ such that the area of the polygon equals ${d_k}^2$. By doing some easy trigonometry within the ...
30
votes
2answers
717 views

Fekete's conjecture on repeated applications of the tangent function

A high-school student named Erna Fekete made a conjecture to me via email three years ago, which I could not answer. I've since lost touch with her. I repeat her interesting conjecture here, in case ...
28
votes
3answers
774 views

$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$

Please help me to solve this integral: $$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx.$$ I managed to calculate an indefinite integral of the left part: $$\int\frac{\cos x}{\sin ...
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?) ...
26
votes
7answers
2k views

Where are the values of the sine function coming from?

On high school, I was taught that I could obtain any sine value with some basic arithmetic on the values of the following image: But I never really understood where these values where coming from, ...
26
votes
4answers
723 views

Geometric explanation of $\sqrt 2 + \sqrt 3 \approx \pi$

Just curious, is there a geometry picture explanation to show that $\sqrt 2 + \sqrt 3 $ is close to $ \pi $?
26
votes
1answer
501 views

$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$

I need help with calculating this integral: $$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$ where ...
26
votes
1answer
1k views
25
votes
8answers
3k views

Reasoning that $ \sin2x=2 \sin x \cos x$

In mathcounts teacher told us to use the formula $ \sin2x=2 \sin x \cos x$. What's the math behind this formula that made it true? Can someone explain?
25
votes
6answers
2k views

If we define $\sin x$ as series, how can we obtain the geometric meaning of $\sin x$?

In Terry Tao's textbook Analysis, he defines $\sin x$ as below: Define rational numbers Define Cauchy sequences of rational numbers, and equivalence of Cauchy sequences Define reals as the ...
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? $$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$
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( ...
23
votes
2answers
338 views

Disproving an “almost true” trigonometric identity

The plausible looking "identity" $$\sin(\frac{\pi}{51})+\cos(\frac{\pi}{74})=\frac{3}{2\sqrt 2}$$ is not true, but it is close indeed: $$LHS=1.0606\color{blue}{598...}$$ ...
22
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 ...
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 ...
22
votes
2answers
518 views

Is the Maclaurin series expansion of $\sin x$ related to the inclusion-exclusion principle?

When I see the alternating signs in the infinite series expansion of $\sin x$, I'm reminded of the inclusion-exclusion principle. Could there be any way to visualize it in such a way? Also, is there ...
21
votes
12answers
4k views

How to solve $4\sin \theta +3\cos \theta = 5$

Another problem that I already wasted hours on. Given $$4\sinθ +3\cosθ = 5$$ Find $$4\cosθ -3\sinθ$$ Help me guys (PS:I'm not that good in maths)
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... :-)
21
votes
8answers
933 views

Mental estimate for tangent of an angle (from $0$ to $90$ degrees)

Does anyone know of a way to estimate the tangent of an angle in their head? Accuracy is not critically important, but within $5%$ percent would probably be good, 10% may be acceptable. I can ...
21
votes
1answer
680 views

A definite integral $\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx$

I need to find a value of this definite integral: $$\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx.$$ Its numeric value is approximately $0.7875720991394284$, and lookups ...
21
votes
2answers
471 views

Is $\{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?

A student asked me the following today : Is $S:= \{\tan(x) : x\in \mathbb{Q}\}$ a group under addition? I am quite perplexed by it. Clearly, the only non-trivial part is to check For any $x, ...
21
votes
2answers
587 views

an integer sum of products of tangents

This question arose from my initial attempts at answering this question. I later found a way to transform the desired sum into a sum of squares of tangents, but before I did, I found numerically that ...
21
votes
3answers
558 views

Evaluating $\int{ \frac{\arctan\sqrt{x^{2}-1}}{\sqrt{x^{2}+x}}} \,dx$

How to integrate? $$\int{ \frac{\arctan\sqrt{x^{2}-1}}{\sqrt{x^{2}+x}}}\, dx$$ I have no idea how to do it. Tried to get some information from wiki, but its too hard :|
20
votes
9answers
3k views

Solve $\cos^2x-\sin^2x= 1$

I am trying to solve for $x$ in the following equation: $$\cos^2x-\sin^2x= 1$$ Given that $\cos^2x+\sin^2x= 1$, is this something I could use to solve it?
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)?
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 $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
19
votes
2answers
3k views

Perfectly centered break of a perfectly aligned pool ball rack

This question is asked on Physics SE and MathOverflow by somebody else. I don't think it belongs there, but rather here (for reasons given there in my comments there; edit: now self-removed). ...
19
votes
2answers
731 views

Which functions satisfy the equation $\,\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y\,$?

Find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that $f(n\pi)=\cos\left(n\pi\right)$ for all ...