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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Tan Binomial formulas from a set S and its k-subsets

Working around, I found some Tan Binomial formulas. Let's $S$ be a set such that: $$ S=\left\{\text{ }\tan ^2\left(\frac{1\pi }{n}\right), \tan^2\left(\frac{2\pi }{n}\right), \tan^2\left(\frac{3\pi }{n}\right)\text{ …
Eddy Khemiri
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For any $x\in\mathbb R$ and any positive integer $n$,$\ $is $\left|\sum_{k=1}^n\frac{\sin{kx}}{k}\right|\le2\sqrt{\pi}$ true?

I'm interested in finding the min of constants $C$ such that $$\left|\sum_{k=1}^n\frac{\sin{kx}}{k}\right|\le C.$$ By using computer, I reached the following expectation: $$\left|\sum_{k=1}^n\frac{\sin{kx}}{k}\right|\le2\sqrt{\pi}$$ for any…
mathlove
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Improving my way of showing $\sin^212^\circ+\sin^221^\circ+\sin^239^\circ+\sin^248^\circ=1+\sin^29^\circ+\sin^218^\circ$

This problem is from 1904 and was given to students studying for the Cambridge and Oxford entry examinations. My solution is presented below, but I am of the opinion that it can be improved. All ideas welcome. Show…
Red Five
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Trigonometric problem using basic trigonometry

If $x$ is a solution of the equation: $$\tan^3 x = \cos^2 x - \sin^2 x$$ Then what is the value of $\tan^2 x$? This is the problem you are supposed to do it just with highschool trigonometry , but i can't manage to do it please help Here are the…
Mxios
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Summation of $\cos A+\cos(A+B)+ \ldots \cos(A+(n-1)B)$

How would I prove the following result? $$\cos A+\cos(A+B)+\ldots +\cos(A+ (n-1)B) = \frac{\sin\left(\frac{nB}{2}\right) \cos\left[A+\frac{(n-1)B}{2}\right]}{\sin \left(\frac{B}{2}\right)}$$
user93343
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Palett Jack trigonometric problem

I have bit of mechanical problem in real life that I need to solve by math. Not sure is this trigonometric problem or something else but here's what I have. I'm not mathematician so don't blame me if I used incorrect markings on illustration above…
Wh1T3h4Ck5
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Proof of $\arctan{2} = \pi/2 -\arctan{1/2}$

I am studying undergraduate complex analysis, and in my Textbook the author claimed that $$\arctan(2)=\frac{\pi}{2}-\arctan\left(\frac{1}{2}\right)$$ when he was doing an example regarding to principle argument. The origin of his claim is…
user2654176
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How to solve a system of trigonometric equations

This question today appeared in my maths olympiad paper: If $\cos x + \cos y + \cos z = \sin x + \sin y + \sin z = 0$, then, prove that $\cos 2x + \cos 2y + \cos 2z = \sin 2x + \sin 2y + \sin 2z = 0$. Can anyone please help me in finding out the…
Vishwesh
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After what interval in degrees or radians do sine, cosine and tangent values repeat?

Between $0$ to $2π$, I have noticed that $\sin x$, $\cos x$ and $\tan x$ values repeat for different values of $x$. For example, $\sin 30 = \sin 150$ What exactly is the interval between two successive values of $x$ such that the value of $\sin x$,…
Niharika
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Finding the range of $\frac{\sin(\alpha+\beta+\gamma)}{\sin\alpha+\sin\beta+\sin\gamma}$, where $\alpha,\beta,\gamma\in\left(0,\frac\pi2\right) $

How do I find the range of : $$ \dfrac{\sin(\alpha +\beta +\gamma )}{\sin\alpha + \sin\beta + \sin\gamma} $$ Where, $$ \alpha , \beta\; and \;\gamma \in \left(0, \frac{\pi}{2}\right) $$ I tried using jensen's inequality on $\alpha, \beta \;and\;…
Bipul Kumar
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Solving for $\tan \theta$ given $\sin \theta/2$

QUESTION: I'm having a hard time figuring this problem out. I've looked through my lectures and cannot find a problem that relates to this one. I do have my identities pulled up in front of me. I'm unsure where to start though. Can someone give me a…
user91539
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What is $\sin 2v $ if $\sin v= 12/13$

I'm having an exam tomorrow and stumbled over an old exam question that says: $$\sin v = \frac{12}{13},\qquad \pi/2 < v <\pi, $$ What is $\sin 2v$? Answer exactly! I've been sitting with this for a few hours, also asking a friend, but we are both…
Anders
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How to prove $\sin^2{7\theta}-\sin^2{4\theta}=\sin{11\theta}\sin{3\theta}$?

I've tried starting from both sides, the furthest I've gone is by starting from the right hand side: $$\begin{align*} \sin{11\theta}\sin{3\theta}&=\sin{(7\theta+4\theta)}\sin{(7\theta-4\theta)}…
edds_7575
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When $\cos(x)+\cos(y)=\sin(x)+\sin(y)=1$, $x$ or $y$ must be a multiple of $2\pi$?

We are given that $\cos x+\cos y=1$ and $\sin x+\sin y=1$. Show that $x+y=(2n+\frac{1}{2})\pi$, when $n$ is an integer. Also show that $x$ or $y$ must be a multiple of $2\pi$. I could get the first result by transforming above two equations to…
Ravindu Kalhara
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Where's my misstep in this trigonometric problem?

$$\begin{align} \tan\alpha+\cot\alpha=\sqrt{ 6 } \\ \tan^6\alpha+\cot^6\alpha= \ ? \end{align}$$ The given answer is $52$, but I got $214$ instead: \begin{align} \tan\alpha+\cot\alpha=\frac{1}{\sin\alpha \cos\alpha}&=\sqrt{ 6 } \\…
L0L1P0P
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