Mathematics Stack Exchange
Mathematics Stack Exchange

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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Show $4 \cos^2{\frac{\pi}{5}} - 2 \cos{\frac{\pi}{5}} -1 = 0$

Show $$4 \cos^2{\frac{\pi}{5}} - 2 \cos{\frac{\pi}{5}} -1 = 0$$ The hint says "note $\sin{\frac{3\pi}{5}} = \sin{\frac{2\pi}{5}}$" and "use double/triple angle or otherwise" So I have $$4 \cos^2{\frac{\pi}{5}} - 2 (2 \cos^2{\frac{\pi}{10}} - 1) -…
Jiew Meng
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Solving the system $\sum \sin = \sum \cos = 0$.

Can we solve the system of equations: $$\sin \alpha + \sin \beta + \sin \gamma = 0$$ $$\cos \alpha + \cos \beta + \cos \gamma = 0$$ ? (i.e. find the possible values of $\alpha, \beta, \gamma$)
vikiiii
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A difficult trigonometry problem

How to prove that $$\left(\sin{\frac{9\pi}{70}}+\sin{\frac{29\pi}{70}}-\sin{\frac{31\pi}{70}}\right)\left(\sin{\frac{\pi}{70}}-\sin{\frac{11\pi}{70}}-\sin{\frac{19\pi}{70}}\right)=\frac{\sqrt{5}-4}{4}?$$ I don't have any idea.
Kong
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How to simpify $\cos x - \sin x$

How does one simplify $$\cos x - \sin x$$ I tried multiplying by $\cos x + \sin x$, but that just gets me $$\cos x - \sin x = \frac{\cos 2x}{\cos x + \sin x}$$ which is worse. Yet wolframalpha gives me $\cos x - \sin x =…
user195518
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Cosine of the sum of two solutions of trigonometric equation $a\cos \theta + b\sin \theta = c$

Question: If $\alpha$ and $\beta$ are the solutions of $a\cos \theta + b\sin \theta = c$, then show that: $$\cos (\alpha + \beta) = \frac{a^2 - b^2}{a^2 + b^2}$$ No idea how to even approach the problem. I tried taking two equations, by substituting…
Gummy bears
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Find $\sec \theta + \tan \theta$.

If $\tan \theta=x-\frac{1}{x}$, find $\sec \theta + \tan \theta$. This was the question ask in my unit test. My Efforts: $\tan^2 \theta=(x-\frac{1}{x})^2$ $\tan^2 \theta= (\frac {x^2-1}{x})^2$ Now we can use identity $\sec^2 \theta= 1 + \tan^2…
RutvikSutaria
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Find the smallest positive number $p$ for which the equation $\cos(p\sin x)=\sin(p \cos x)$ has a solution $x\in[0,2\pi].$

Find the smallest positive number $p$ for which the equation $\cos(p\sin{x})=\sin(p\cos{x})$ has a solution $x$ belonging $[0,2\pi]$. I am not able to solve this problem. Please help me.
geek101
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Simplify $2 \sin(x) \cos(7x) + \sin(6x)$

I was doing a problem and in my chain of computations I arrived at a seemingly complicated function $$2 \sin(x) \cos(7x) + \sin(6x)$$ However, I typed it into Wolfram and was surprised to find $$2 \sin(x) \cos(7x) + \sin(6x) = \sin(8x)$$ Is…
Mark
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Find the Value of Trigonometric Expression

Given $$\begin{align} \frac{\cos \alpha}{\cos \beta}+\frac{\sin \alpha}{\sin \beta}=-1 \end{align} \tag{1}$$ Find the value of $$\begin{align} \frac{\cos^3 \beta}{\cos \alpha}+\frac{\sin ^3\beta}{\sin \alpha} \end{align} \tag{2} $$ I Tried like…
Ekaveera Gouribhatla
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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: $$f(x)=\cos(x/3)+\cos(x/4).$$
Jaydeep
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Trig Identity Proof of Tan(x) + Tan(y)

I am trying to prove the identity below to help with the simplification of another function that I'm investigating as it doesn't appear to be a standard trig identity. $$ \tan\left(x\right) + \tan\left( y \right) = \frac{{\sin\left( {x + y}…
ASBO Allstar
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Help me prove: sin(A+B) = sinA cosB + cosA sinB

Can you help me prove that: sin(A+B) = sinA cosB + cosA sinB? Thanks!
Gauss
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Prove $\prod_{i=1}^{n-1} \sin(i\pi/n) = 2^{1-n} n$ without complex functions.

Note that $i$ here refers to indexing variable, not $\sqrt{-1}$. $$\prod_{i=1}^{n-1} \sin\left(\frac{i \pi}{n}\right) = 2^{1-n} n$$ This formula was used here to give an 'elementary' proof of product of diagonals = N. Mathworld is the only place I…
genepeer
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Find the minimum of $\displaystyle \frac{1}{\sin^2(\angle A)} + \frac{1}{\sin^2(\angle B)} + \frac{1}{\sin^2(\angle C)}$

Is it possible to find the minimum value of $E$ where $$E = \frac{1}{\sin^2(\angle A)} + \frac{1}{\sin^2(\angle B)} + \frac{1}{\sin^2(\angle C)}$$for any $\triangle ABC$. I've got the feeling that $\min(E) = 4$ and that the critical value occurs…
Mick
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Maximum and Minimum value of an inverse function

Find the maximum and minimum value of $\arcsin \left(x\right)^3+\arccos \left(x\right)^3$. given that $-1\le x\le 1$ I have solved the problem but i am just curious to know if there are any other ways to solve this particular problem other than…
Adesh
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