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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Proving a trigonometric identity

How can one prove the validity of this trigonometric identity? \begin{equation} 2\arccos\sqrt{x} = \frac{\pi }{2}-\arcsin(2x-1) \end{equation}
liberias
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Proof of the angle sum identity for $\sin$

$$\sin(a+b) = \sin(a) \cos(b) + \cos(a) \sin(b)$$ How can I prove this statement?
Kuba
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$\sin n$ having a closed form expression with no complex terms?

[All angles are in degrees] I have heard that we cannot express $\sin 1$ in closed form with no complex terms. However, I know that we can derive $\sin 18$ by solving $\cos 3x = \sin 2x$. Thus we can get trig values for $72$ degrees and then get to…
Sawarnik
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Find where line intersects sine function

Here we have sin((x+1)*(pi/2)): Now let's say we have a point at (x=2, y=1)... And we draw a line to the origin (x=0, y=0)... If we create a point where the line intersects the sine function, we get a new point. Is there a way to find out the…
programmer
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Trigonometric system

I would like to solve: $ x +y+z=\frac{11\pi}{6} $ $ \sin(x)+\sin(y)+\sin(z)= \frac{\sqrt{3}}{2} $ $ \cos(x)+\cos(y)+\cos(z)=\frac{1}{2} $ After eliminating $ z $ I get: $ 2\sin(x)+2\sin(y)-\cos(x+y)-\sqrt{3}\sin(x+y)=\sqrt{3}\tag{1}$ $…
Chon
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Solve the equation $\tan \theta = 2\sin \theta$.

Solve the equation $\tan \theta = 2\sin \theta$. What I did was rewrite it to the form $$\sin \theta = 2 \sin \theta \cos \theta$$ You'll get $$\sin \theta = \sin\ 2 \theta.$$ How am I supposed to solve this when I have $\sin$ on both sides? My…
Phaptitude
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Why does there seem to be so much error in the laws of sines and cosines?

I've been computing the angles of a triangle with sides a = 17, b = 6 and c = 15 using the law of cosines to find the first angle and then the law of sines to find the other 2. I follow the convention of naming the angles opposite these sides A, B…
George Tomlinson
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Calculating point on a circle, given an offset?

I have what seemed like a very simple issue, but I just cannot figure it out. I have the following circles around a common point: The Green and Blue circles represent circles that orbit the center point. I have been able to calculate the…
George
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Trigonometry: Solve $(1-\cos\alpha)^2 + \sin^2\alpha = d^2$ for $\alpha$

My next step in implementing my algorithm in Java is following. It is quite difficult to explain, but I know what I need. I have this equation: Given: d Asked: $\alpha$ $$(1-\cos\alpha)^2 + \sin^2\alpha = d^2 $$ Which I "simplified" to this, using…
Martijn Courteaux
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Coincidental Trigonometric Identity for Two Particular Values

I noticed that $$\sin(a+b)\sin(a-b) = \cos a \cos b\qquad (1)$$ when $$(a,b)=\left(\frac{2\pi}{5},\frac{\pi}{3}\right)$$ Is there an underlying reason for this coincidence? Concretely, I would like to answer these questions: Does the solution set…
pre-kidney
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Solve, $(\cos x-\sin x)(2\tan x+\sec x)+2=0$

I solved this problem by converting $\sin x,\cos x,\sec x$ all in terms of $\tan(x/2)$ which gives a biquadratic. I found $2$ roots of the biquadratic by hit and trial i.e $\tan^2(x/2) = 1/3$ but the calculations were really lengthy. I had also…
Anshu Gupta
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Simplifying $\frac{\sin(3x)+\sin(4x)+\sin(5x)}{\sin(x)+\sin(2x)+\sin(3x)}$

To simplify: $$\frac{\sin(3x)+\sin(4x)+\sin(5x)}{\sin(x)+\sin(2x)+\sin(3x)}.$$ I know that in the denominator, $\sin(2x)$ can be re-written as $2\sin x\cos x$ using the double-angle identity. I also know that for the numerator…
jlemtryung
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Period of Trigonometric Functions

I have always been taught that in the scenario of a Sine,Tan,Cos function of $f(x) = a\sin b(x+c) +d$, the period of the sine and cos functions $= \dfrac{2\pi}{b}$, and the period for the tan function $= \dfrac{\pi}{b}$ I don't see how this would…
Harrison
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Proving a fact: $\tan(6^{\circ}) \tan(42^{\circ})= \tan(12^{\circ})\tan(24^{\circ})$

Prove that $\tan(6^\circ)\tan(42^\circ) = \tan(12^\circ) \tan(24^\circ)$. I don't know how to approach this problem. One approach might be to note that $42-6= 24+12$, and then apply the identities for $\tan(A+B)$ and $\tan(A-B)$, but it just makes…
HelloWorld
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How do you solve $\cos (3x) = \cos (x)$?

I want to solve this problem: $\cos(3x) = \cos(x)$ and I’m stuck. I have tried to rewrite it to $4 \cos^3(x) - 3 \cos(x) = \cos (x)$ and then solve it. Add $3 \cos (x)$ to both sides and then divide by $4$. And then I have this: $\cos^3(x) = \cos…
Ridertvis
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