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.

29665 questions
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Solve this Trigonometric equation

I am not quite good at maths, so can you help me ? $$\tan x + 2 \cot x - 3=0$$
Bee
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Funny Trig Math Puzzle

This is a challenging puzzle I heard from my little brother. For some $n$ and $x$, $\sum_{k=1}^n \sin^{2k}(x) = 2013$. Is it possible to deduce $$\sum_{k=1}^n \cos^{2k}(x) \text{ ?}$$ Edit: I've just noticed something which now seems obvious to…
Mark
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Understanding arcsin inequalities

Assuming this is correct: $\arcsin x \neq 0$ $\sin(0) \neq x$ $0\neq x$ Following the same logic, why is this incorrect? $\arcsin(x+\frac{1}{3}) \geq 0$ $\sin (0) \geq x+\frac{1}{3}$ $0 \geq x + \frac{1}{3}$ $x \leq -\frac{1}{3}$ $x\in(-\infty;…
weno
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Convert $\mathrm{i}^\pi$ into trigonometric form

How can I convert the complex number $\mathrm i^\pi$ to trigonometric form? I usually do these steps: take $ Z = a + b\mathrm i $ form, find $ r = \sqrt{a^2 + b^2} $, $ \cos(\phi) = a / r, \sin(\phi) = b / r $, find $ \phi $ from the above 2…
user3132457
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Trigonometry with multiple angle and exact value of $\tan\pi/5$

By considering the equation $\tan5\theta=0$, show that the exact value of $\tan\pi/5$ is $\sqrt{5-2\sqrt{5}}$. Do I need to evaluate the multiple angle for $\tan5\theta=0$?
bbr4in
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Trigonometry problem concerning a point outside an isosceles triangle

I have a hard time solving the following problem: Given an Isosceles triangle $\triangle ABC$ where $AC = BC$, with $\angle ACB = 50^{\circ}$ let $M$ be a point outside the triangle $\triangle ABC$ but within the angle $\angle BAC$. If $\angle AMB =…
ogv
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How to prove that $\sin(\sqrt{x})$ is not periodic?

How to prove that $\sin(\sqrt{x})$ is not periodic? THe definition of a periodic function is $f(x+P)=f(x)$. So I assume that $\sin(\sqrt{x+P})=\sin(\sqrt{x})$. This is equivalent to $\sin(\sqrt{x+P})-\sin(\sqrt{x})=0$. This implies…
James Warthington
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How to solve this equation $\sin 5x = \sin (x + \frac{\pi}{3})$

Could you give a hint how to solve this equation $\sin 5x=\sin (x + \frac{\pi}{3})$? I tried to change $\sin 5x$ in function of $\sin x$ and $\cos x$, but I wasn't able to go further.
user23505
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Evaluating $\frac{1}{\sin(2x)} + \frac{1}{\sin(4x)} + \frac{1}{\sin(8x)} + \frac{1}{\sin(16x)}$

Evaluate $$\dfrac{1}{\sin(2x)} + \dfrac{1}{\sin(4x)} + \dfrac{1}{\sin(8x)} + \dfrac{1}{\sin(16x)}$$ It would be tough for us to solve it using trigonometric identities. There should be strictly an easy trick to proceed. Rewriting and using…
Busi
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Solve the equation $\frac{\sqrt {3}}{2}\sin(x) -\cos x=\cos^2x$

Solve the equation $\frac{\sqrt {3}}{2}\sin x-\cos x=\cos^2x$ My approach $\cos^2x=1-\sin^2x $ $\frac{\sqrt {3}}{2}\sin x-\cos^2x =\cos x$ $\frac{\sqrt {3}}{2}\sin x+\sin^2x -1=\cos x$ $\sqrt {3}\sin x+2\sin^2x-2=2\cos x$ Though the equation comes…
Samar Imam Zaidi
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Powers of trigonometric functions

Are formulas known for non-integer powers of trigonometric functions, analgolous to the power reduction identities? Like if you had a root of sine, could you express it in as a finite summation of sine and cosine functions with different inputs?
Jacob Lewis
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If $\sin x +\sin 2x + \sin 3x = \sin y\:$ and $\:\cos x + \cos 2x + \cos 3x =\cos y$,

If $\sin x +\sin 2x + \sin 3x = \sin y\:$ and $\:\cos x + \cos 2x + \cos 3x =\cos y$, then $x$ is equal to (a) $y$ (b) $y/2$ (c) $2y$ (d) $y/6$ I expanded the first equation to reach $2\sin x(2+\cos x-2\sin x)= \sin y$, but I doubt it leads me…
ibuprofen
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Finding $\sin 2x+\sin 4x+\cdots+\sin 22x$

I have to find this sum: $$S= \sin(2x)+\sin(4x)+\sin(6x)+\cdots+\sin(22x).$$ I tried to multiply the $S$ by $i$ then add a so called "$T$" where $$T = \cos 2x + \cos 4x +\cdots+\cos 22x.$$ From here I obtained: $$T+iS=(\cos x+i\sin x)^2+(\cos…
Robert Maracine
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Find value of $\tan\big(\frac{\pi}{25}\big)\cdot \tan\big(\frac{2\pi}{25}\big)\cdots\tan\big(\frac{12\pi}{25}\big)$

Find value of: $$\displaystyle \tan\bigg(\frac{\pi}{25}\bigg)\cdot \tan\bigg(\frac{2\pi}{25}\bigg)\cdot \tan\bigg(\frac{3\pi}{25}\bigg)\cdots\cdots \tan\bigg(\frac{12\pi}{25}\bigg)$$ The solution I tried: Assume $$P = …
jacky
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How to fit a sinusoid to 2 points and their gradients

Given the sinusoidal function $$f(x) = a \cos(n x + b) + c,$$ if I know $f(x_1)$, $f(x_2)$, $f'(x_1)$ and $f'(x_2)$ is it possible to determine $a, b, c$ and $n$, with $x \in [0,\tfrac{2\pi}{n})$ Edit: put bounds on $x$ so that only one complete…
geometrikal
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