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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Prove the trigonometric identity $(35)$

Prove that \begin{equation} \prod_{k=1}^{\lfloor (n-1)/2 \rfloor}\tan \left(\frac{k \pi}{n}\right)= \left\{ \begin{aligned} \sqrt{n} \space \space \text{for $n$ odd}\\ \\ \ 1 \space \space \text{for $n$ even}\\ …
user 1591719
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Solve $\cos(\theta) + \sin(\theta) = x$ for known $x$, unknown $\theta$?

After looking at the list of trigonometric identities, I can't seem to find a way to solve this. Is it solvable? $$\cos(\theta) + \sin(\theta) = x.$$ What if I added another equation to the problem: $$-\sin(\theta) + \cos(\theta) = y,$$ where…
levesque
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Evaluate $\tan^{2}(20^{\circ}) + \tan^{2}(40^{\circ}) + \tan^{2}(80^{\circ})$

Evaluate $\tan^{2}(20^{\circ}) + \tan^{2}(40^{\circ}) + \tan^{2}(80^{\circ})$. Can anyone help me with this? Thank You!
JSCB
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Simplifying $\cos(\arcsin x)$?

I remember reading somewhere that we can simplify $\cos(\arcsin x)$ and $\sin(\arccos x)$ in terms of a polynomial by making the substitution $m=\arcsin x$ or $m=\arccos x$ (respectively), then constructing a right angle triangle with appropriate…
CivilSigma
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How do we prove $\cos(\pi/5) - \cos(2\pi/5) = 0.5$ ?.

How do we prove $\cos(\pi/5) - \cos(2\pi/5) = 0.5$ without using a calculator. Related question: how do we prove that $\cos(\pi/5)\cos(2\pi/5) = 0.25$, also without using a calculator
ranadheer
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How to calculate $\cos(6^\circ)$?

Do you know any method to calculate $\cos(6^\circ)$ ? I tried lots of trigonometric equations, but not found any suitable one for this problem.
Mahdi Khosravi
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Isn't the book wrongly taking $\sin^{-1}$ on both sides here?

Question: If $\sin(\pi\cos\theta)=\cos(\pi\sin\theta)$, then show that $\theta=\pm\frac{1}{2}\sin^{-1}\frac{3}{4}$. My book's solution: $$\sin(\pi\cos\theta)=\cos(\pi\sin\theta)$$ $$\sin(\pi\cos\theta)=\sin(\frac{\pi}{2}\pm\pi\sin\theta)\…
tryingtobeastoic
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Solve $\sin^{3}x+\cos^{3}x=1$

Solve for $x\\ \sin^{3}x+\cos^{3}x=1$ $\sin^{3}x+\cos^{3}x=1\\(\sin x+\cos x)(\sin^{2}x-\sin x\cdot\cos x+\cos^{2}x)=1\\(\sin x+\cos x)(1-\sin x\cdot\cos x)=1$ What should I do next?
user593646
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Proving :$\arctan(1)+\arctan(2)+ \arctan(3)=\pi$

Possible Duplicate: Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$? How to prove $$\arctan(1)+\arctan(2)+ \arctan(3)=\pi$$
hin
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can we have a triangle with sides $1, x$ and $x^2$?

Can we have a triangle with sides $1$, $x$ and $x^2$? And what could $x$ be? I try to approach this question by making 3 inequalities. $1+x>x^2$, $1+x^2>x$, $x^2+x>1$ and they come with different quadratic inequalities $x^2-x-1<0$ (solution is…
user51658
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Can someone confirm my method and answer for this trig problem?

Find the exact value of: $$\cos 1^{\circ}+\cos 2^{\circ}+\cos 3^{\circ}+ \ldots +\cos 358^{\circ}+\cos 359^{\circ}$$ I got $0$ as I did this by assigning either a positive or negative $x$ variable for each quadrant. Is this method valid and my…
Sarah Rubenstein
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Calculating :$((\sqrt{3} + \tan (1^\circ)).((\sqrt{3} +\tan(2^\circ))...((\sqrt{3}+\tan(29^\circ))$

What is the easiest way to calculate : $$(\sqrt{3} + \tan (1^\circ)).((\sqrt{3} +\tan(2^\circ))...((\sqrt{3}+\tan(29^\circ)) $$
sib
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Is there a formula for the cosine of 1/5 of an angle?

I'm trying to find a formula for $\cos \frac{x}{5} $ as follow. By the elementary trigonometric identity, $\cos 5x = 16\cos^5 x - 20\cos^3 x + 5\cos x$ By putting $x= \frac{x}{5}$ , and using the change of variable $ y = \cos \frac{x}{5} $, we get…
Jeremy
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Resolve $\cos(3x)= \cos(2x)$

I have to solve $$\cos(3x)= \cos(2x)$$ I found how to express both of them for $x$ only and now I have $$4\cos^3(x)-3\cos(x)=2\cos^2(x)-1$$ What do I do now?
Rosalie Landry-Ouellet
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Using the unit circle to prove the double angle formulas for sine and cosine?

How do you use the unit circle to prove the double angle formulas for sine and cosine?
user8210
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