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

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12
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5answers
6k views

Why is it that when proving trig identities, one must work both sides independently?

Suppose that you have to prove the trig identity: $$\frac{\sin\theta - \sin^3\theta}{\cos^2\theta}=\sin\theta$$ I have always been told that I should manipulate the left and right sides of the ...
10
votes
3answers
14k views

Equation of angle bisector, given the equations of two lines in 2D

I have two lines in 2D expressed with general equation (or implicit equation): First line: $a_1x+b_1y=c_1 \qquad(1)$ Second line: $a_2x+b_2y=c_2 \qquad(2)$ If the two lines are intersecting I will ...
6
votes
1answer
725 views

Continued fraction for $\tan(nx)$

I found this beautiful continued fraction expansion of $\tan(nx)$, $n$ being a positive integer, online but I don't remember the source now: $\displaystyle \tan(nx) = \cfrac{n\tan x}{1 -\cfrac{(n^{2} ...
5
votes
3answers
227 views

Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$

Calculate the definite integral $$ I=\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x}\;dx $$ given that $a>b>0$ My Attempt: If we replace $x$ by $C$, then $$ I = ...
5
votes
5answers
494 views

Evaluate $\int_0^{\frac{\pi}{2}} \ln(1+\cos x)\, dx$

Find the value of $\displaystyle \int_0^{\frac{\pi}{2}} \ln(1+\cos x) $ I tried to put $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but I am unable to proceed further. I think the following integral can be ...
5
votes
3answers
2k views

How do I measure distance on a globe?

I have a $3$-D sphere of radius $R$, centered at the origin. $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ are two points on the sphere. The Euclidean distance is easy to calculate, but what if I were to ...
3
votes
2answers
151 views

The trigonometric solution to the solvable DeMoivre quintic?

Using the relations for the Rogers-Ramanujan cfrac described in this post, $$\frac{1}{r}-r = x$$ $$\frac{1}{r^5}-r^5 = y$$ and eliminating $r$ yields, $$x^5+5x^3+5x = y$$ This is the case $a=1$ ...
3
votes
6answers
21k views

$\cos(\arcsin(x)) = \sqrt{1 - x^2}$. How?

How does that bit work? How is $$\cos(\arcsin(x)) = \sin(\arccos(x)) = \sqrt{1 - x^2}$$
2
votes
4answers
224 views

If $\alpha = \frac{2\pi}{7}$ then the find the value of $\tan\alpha .\tan2\alpha +\tan2\alpha \tan4\alpha +\tan4\alpha \tan\alpha.$

If $\alpha = \frac{2\pi}{7}$ then the find the value of $\tan\alpha .\tan2\alpha +\tan2\alpha \tan4\alpha +\tan4\alpha \tan\alpha$ My 1st approach : $\tan(\alpha +2\alpha +4\alpha) = ...
17
votes
3answers
461 views

$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ$

How can I find the following product using elementary trigonometry? $$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ.$$ I have tried using a substitution, but nothing ...
11
votes
6answers
1k views

Complex roots of $z^3 + \bar{z} = 0$

I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre. Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
7
votes
2answers
149 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 $$\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$$
3
votes
4answers
437 views

Find the value of $\textrm{cosec}^2\left(\frac\pi7\right) +\textrm{cosec}^2\left(\frac{2\pi}7\right)+\textrm{cosec}^2\left(\frac{4\pi}7\right)$

What is the value of $$\textrm{cosec}^2\left(\frac\pi7\right) +\textrm{cosec}^2\left(\frac{2\pi}7\right)+\textrm{cosec}^2\left(\frac{4\pi}7\right) \qquad\qquad ? $$ I tried to write ...
0
votes
3answers
163 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
11
votes
5answers
1k views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: ...
10
votes
4answers
1k views

Is there a more efficient method of trig mastery than rote memorization?

I would like to get alot better at trig than I am. What is the best/most efficient method? Thanks much in advance Joe
6
votes
2answers
293 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 ...
6
votes
6answers
986 views

Verify the identity: $\tan^{-1} x +\tan^{-1} (1/x) = \pi /2$

Verify the identity: $\tan^{-1} x + \tan^{-1} (1/x) = \frac\pi 2, x > 0$ $$\alpha= \tan^{-1} x$$ $$\beta = \tan^{-1} (1/x)$$ $$\tan \alpha = x$$ $$\tan \beta = 1/x$$ $$\tan^{-1}[\tan(\alpha + ...
5
votes
1answer
180 views

Finite Series - reciprocals of sines

Find the sum of the finite series $$\sum _{k=1}^{k=89} \frac{1}{\sin(k^{\circ})\sin((k+1)^{\circ})}$$ This problem was asked in a test in my school. The answer seems to be ...
5
votes
1answer
770 views

The only two rational values for cosine and their connection to the Kummer Rings

I am trying to learn about Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers) The only $\theta\in [0,\frac{\pi}{2}]$ which are rational ...
4
votes
3answers
84 views

Why is the range of inverse trigonometric functions defined in this way?

My question is really simple, why is the range of the $\sin^{-1}(x)$, $\cos^{-1}(x)$ and $\tan^{-1}(x)$ defined as $[-\pi/2,\pi/2]$, $[0,\pi]$ and $[-\pi/2,\pi/2]$ respectively? Is there some ...
4
votes
4answers
863 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
1answer
112 views

geometric proof of $2\cos{A}\cos{B}=\cos{(A+B)}+\cos{(A-B)}$

I have seen geometric proof of identities $$\cos{(A+B)}=\cos{A}\cos{B}-\sin{A}\sin{B}$$ and $$\cos{(A-B)}=\cos{A}\cos{B}+\sin{A}\sin{B}$$ By adding two equation, ...
3
votes
2answers
985 views

How to prove $\cos 36^{\circ} = (1+ \sqrt 5)/4$?

Given $4 \cos^2 x -2\cos x -1 = 0$. Use this to show that $\cos 36^{\circ} = (1+ \sqrt 5)/4$, $\cos 72^{\circ} = (-1+\sqrt 5)/4$ Your help is greatly appreciated! Thanks
3
votes
3answers
2k views

How can I find the derivative of $y = \sin(\arctan x) + \tan(\arcsin x)$?

My question is, how can I solve the following derivative question? $$y = \sin(\arctan x) + \tan(\arcsin x)$$ Thanks in advance
2
votes
3answers
1k views

Non-trigonometric Continuous Periodic Functions

I've seen lots of examples of periodic functions, but they all have one thing in common: They all involve at least one trigonometric term (e.g. $\sin\theta$, $\cos\theta$, etc.). My question is ...
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} ...
4
votes
2answers
205 views

Calculating certain functions if only certain buttons on a calculator are permitted

A calculator is broken. The only keys that work are $\sin, \cos, \tan, \cot, \arcsin, \arccos$, and $\arctan$ buttons. The original display is $0$. In this problem, we will prove that ...
4
votes
4answers
2k views

Linear combinations of sine and cosine

If you take a linear combination of the cosine and sine function, then the result is again a sinusoid, but with a new amplitude and phase shift. $$a \cos(\theta) + b \sin(\theta) = A \cos(\theta + ...
3
votes
2answers
4k views

How to find the period of the sum of two trigonometric functions

I want to know if there exists a general method to find the period of the sum of two periodic trigonometric function. Example: $$f(x)=\cos(x/3)+\cos(x/4).$$
2
votes
6answers
181 views

Evaluating and proving $\lim_{x\to\infty}\frac{\sin x}x$

I've just started learning about limits. Why can we say $$ \lim_{x\rightarrow \infty} \frac{\sin x}{x} = 0 $$ even though $\lim_{x\rightarrow \infty} \sin x$ does not exist? It seems like the fact ...
2
votes
3answers
73 views

meaning of powers on trig functions

I always forget this, when a trig function has an exponent does that mean multiply itself or apply itself to the result recursivly? e.g. does $\sin(x)^2=\sin(x)\sin(x)$ or $=\sin(\sin(x))$? What ...
2
votes
3answers
1k views

Prove that $x - \frac{x^3}{3!} < \sin x < x$ for all $x>0$

Prove that $x - \frac{x^3}{3!} < \sin(x) < x$ for all $x>0$ This should be fairly straightforward but the proof seems to be alluding me. I want to show $x - \frac{x^3}{3!} < ...
2
votes
3answers
125 views

Fractional Trigonometric Integrands

$$∫\frac{a\sin x+b\cos x+c}{d\sin x+e\cos x+f}dx$$ $$∫\frac{a\sin x+b\cos x}{c\sin x+d\cos x}dx$$ $$∫\frac{dx}{a\sin x+\cos x}$$ What are the relations between the numerator in the denominator, and ...
53
votes
2answers
2k 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} ...
38
votes
3answers
813 views

Calculate the following infinite sum in a closed form $\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$?

Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction. $$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
49
votes
4answers
2k views

Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$

I'm interested in integrals of the form $$I(a,b)=\int_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx,\color{#808080}{\text{ for ...
47
votes
5answers
4k 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) ...
42
votes
1answer
988 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 ...
18
votes
1answer
325 views

Some trigo identities

I aacidently found the following: $$\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}-\sin\frac{6\pi}{7}=\frac{\sqrt{7}}{2}$$ ...
17
votes
3answers
941 views

How prove this $\tan{\frac{2\pi}{13}}+4\sin{\frac{6\pi}{13}}=\sqrt{13+2\sqrt{13}}$

Nice Question: show that: The follow nice trigonometry $$\tan{\dfrac{2\pi}{13}}+4\sin{\dfrac{6\pi}{13}}=\sqrt{13+2\sqrt{13}}$$ This problem I have ugly solution,maybe someone have nice ...
17
votes
4answers
1k views

Showing $\tan\frac{2\pi}{13}\tan\frac{5\pi}{13}\tan\frac{6\pi}{13}=\sqrt{65+18\sqrt{13}}$

How can one show : $$\tan\frac{2\pi}{13}\tan\frac{5\pi}{13}\tan\frac{6\pi}{13}=\sqrt{65+18\sqrt{13}}?$$
13
votes
5answers
491 views

How to evaluate $\int_0^1 (\arctan x)^2 \ln(\frac{1+x^2}{2x^2}) dx$

Evaluate $$ \int_{0}^{1} \arctan^{2}\left(\, x\,\right) \ln\left(\, 1 + x^{2} \over 2x^{2}\,\right)\,{\rm d}x $$ I substituted $x \equiv \tan\left(\,\theta\,\right)$ and got $$ ...
9
votes
9answers
13k views

Good book recommendations on trigonometry

I need to find a good book on trigonometry, I was using trigonometry demystified but I got sad when I read this line: Now that you know how the circular functions are defined, you might wonder how ...
19
votes
4answers
283 views

A triangle determinant that is always zero

How do we prove, without actually expanding, that $$\begin{vmatrix} \sin {2A}& \sin {C}& \sin {B}\\ \sin{C}& \sin{2B}& \sin {A}\\ \sin{B}& \sin{A}& \sin{2C} \end{vmatrix}=0$$ ...
18
votes
5answers
6k views

Do “imaginary” and “complex” angles exist?

During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting: $ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $ $ \sin ...
18
votes
3answers
2k views

Sine Approximation of Bhaskara

An Indian mathematician, Bhaskara I, gave the following amazing approximation of the sine (I checked the graph and some values, and the approximation is truly impressive.) $$\sin x \approx ...
16
votes
0answers
309 views

Simplify the sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes $\le N^2$

I was looking for a closed form but it seemed too difficult. Now I'm seeking help to simplify this sum. The 50 bounty points or more will be awarded for any meaningful simplification of this sum. I ...
21
votes
8answers
5k views

Rapid approximation of $\tanh(x)$

This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to ...
14
votes
9answers
7k views

How Can One Prove $\cos(\pi/7) + \cos(3 \pi/7) + \cos(5 \pi/7) = 1/2$

Reference: http://xkcd.com/1047/ We tried various different trigonometric identities. Still no luck. Geometric interpretation would be also welcome. EDIT: Very good answers, I'm clearly impressed. ...