Mathematics Stack Exchange
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Questions tagged [trigonometry]

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

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles.

Trigonometry is most simply associated with planar right-angle triangles. The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles.

One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. Trigonometry on surfaces of negative curvature is part of hyperbolic geometry.

Trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles.

See Wikipedia's list of trigonometric identities.

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How were trigonometrical functions of $\dfrac{2\pi}{17}$ calculated?

I know they were calculated by Gauss, but how? Is there a method for calculating them?
dot dot
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Find $\sin^3 a + \cos^3 a$, if $\sin a + \cos a$ is known

Given that $\sin \phi +\cos \phi =1.2$, find $\sin^3\phi + \cos^3\phi$. My work so far: (I am replacing $\phi$ with the variable a for this) $\sin^3 a + 3\sin^2 a *\cos a + 3\sin a *\cos^2 a + \cos^3 a = 1.728$. (This comes from cubing the…
Mathy Person
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$\sin 4x +\sqrt{3} \sin 3 x + \sin 2 x=0$

This question is from a 2012 VMK entrance exam I was trying to solve it first by expanding $\sin 4 x = 2 \sin 2 x \cos 2x$, then by noticing that if divided by 2, one can get, e.g. $ \frac{\sqrt{3}}{2} \cos 3 x = \sin \frac{\pi}{6} \cos 3 x$ and…
Alex
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math question angle of elevation

A tree is $x$ meters high. The angle of elevation of its top from a point $P$ on the ground is 23 degrees. From another point $Q$, 10 meters from $P$ and in line with $P$ and the foot of the tree, the angle of elevation is 32 degrees. Find $x$.…
Asad Rafiq
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Show that $\sin 10^\circ$ is irrational

So, this is the problem I am working on. Show that $\sin 10^\circ$ is irrational. The solution to the problem is $$1/2 = \sin 30^\circ = 3 \sin 10^\circ - 4\sin^3 10^\circ .$$ Let $$x = 2\sin 10^\circ.$$ Then we have, $$x^3 - 3x + 1 = 0.$$ And,…
user136422
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Verify the identity: $\tan^{-1} x +\tan^{-1} (1/x) = \pi /2$

Verify the identity: $\tan^{-1} x + \tan^{-1} (1/x) = \frac\pi 2, x > 0$ $$\alpha= \tan^{-1} x$$ $$\beta = \tan^{-1} (1/x)$$ $$\tan \alpha = x$$ $$\tan \beta = 1/x$$ $$\tan^{-1}[\tan(\alpha + \beta)]$$ $$\tan^{-1}\left [{\tan\alpha + \tan\beta\over…
KKendall
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Solve equation $ \cos x+\sin x=0$

I'm trying to solve an equation here but unfortunately I can't. The equation: $$ \cos x + \sin x = 0 $$ I'm trying to solve this by replacing $\cos x$ with $(1-t^2)/(1+t^2)$ and $\sin x$ with $2t/(1+t^2), t=\tan x/2, \ $ but I can't get the right…
xxx
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How to find this value: $\sum_{i_1=1}^n \sum_{i_2=1}^n \cdots\sum_{i_k=1}^n \cos\frac{k(i^k_1+i^k_2+\cdots+i^k_k)}{n}$

Question: Find this value $$\sum_{i_{1} = 1}^{n}\sum_{i_{2} = 1}^{n}\cdots\sum_{i_{k} = 1}^{n} \cos\left(k\left[\vphantom{\Large A}\, i_{1}^{k} + i_{2}^{k} + \cdots +i_{k}^{k}\,\right] \over n\right) $$ This is an interesting problem, My try:…
math110
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Hard contest type trigonometry proof

Suppose that real numbers $x, y, z$ satisfy: $$\frac{\cos x + \cos y + \cos z}{\cos(x + y + z)} = \frac{\sin x + \sin y + \sin z}{\sin (x + y + z )} = p$$ Then prove that: $$\cos (x + y) + \cos (y + z ) + \cos (x + z) = p$$ I am not even getting…
Sawarnik
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Problem : If $\sin^2\theta = \frac{x^2+y^2+1}{2x}$ , $x$ must be ....

Problem : If $\sin^2\theta = \frac{x^2+y^2+1}{2x}$ , $x$ must be (a) $1$ (b) $-2$ (c) $-3$ (d) $ 2$ My approach : Since $0 \leq \sin^2\theta \leq 1$ $\Rightarrow 0 \leq \frac{x^2+y^2+1}{2x} \leq 1 $ $\Rightarrow x^2 +y^2+1 -2x \leq…
Sachin
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Find $\cos(x+y)$ if $\sin(x)+\sin(y)= a$ and $\cos(x)+\cos(y)= b$

Find $\cos(x+y)$ if $\sin(x)+\sin(y)= a$ and $\cos(x)+\cos(y)= b$.
jay
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Why does $\tan10^\circ+\tan20^\circ+\tan50^\circ$ equal $\tan60^\circ$?

My calculator says that $\tan10^\circ+\tan20^\circ+\tan50^\circ=\tan60^\circ$, and this is confirmed to 15 decimal places by a more precise online calculator. So it looks plausible. However, I have not managed to prove the result, despite its…
John Bentin
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If $\cos^4 \theta −\sin^4 \theta = x$. Find $\cos^6 \theta − \sin^6 \theta $ in terms of $x$.

Given $\cos^4 \theta −\sin^4 \theta = x$ , I've to find the value of $\cos^6 \theta − \sin^6 \theta $ . Here is what I did: $\cos^4 \theta −\sin^4 \theta = x$. ($\cos^2 \theta −\sin^2 \theta)(\cos^2 \theta +\sin^2 \theta) = x$ Thus ($\cos^2…
A Googler
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show this indentity $\sum_{k=1}^{2n-1}\frac{\sin{\frac{k^2\pi}{2n}}}{\sin{\frac{k\pi}{2n}}}=n$

let $n$ be postive integers.show that $$\sum_{k=1}^{2n-1}\dfrac{\sin{\frac{k^2\pi}{2n}}}{\sin{\frac{k\pi}{2n}}}=n$$ Try:I can show $n$ is smaller number. let $LHS=f(n)$.when $n=1$ it is clear $$f(1)=\dfrac{\sin{\pi/2}}{\sin{\pi/2}}=1$$ when $n=2$…
math110
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A hard 'if and only if' trigonometric identity proof

Prove $$ \frac{-2+2\tan A+2\cos B\cdot\sin B+\cot^2 A\cdot({\sec^4A-\operatorname{cosec}^2A-2)}}{2+\tan^2A-2\sin^2A} =(\sin A+\cos A)^2 $$ if and only if B is the double angle of A, or $B=2A+2k\pi$, $k=0,1,2,3...$ Advice is welcome as to improve…
Y-dog
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