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.

29665 questions
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Find all solutions of the equation $\sin x-\frac{6}{x}=0$ for $x\in[0,12\pi]$

I need to solve the equation $\sin x-\frac{6}{x}=0$ for $x\in[0,12\pi]$. I tried substituting $\sin x=\frac{e^{ix}-e^{-ix}}{2i}$ and solving the quadratic but it did not lead to anything. It has 10 solutions but I do not know how to get the exact…
Adit Jain
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Inverse trig function equation

How would you suggest I go about solving this question? I've been thinking about it for ages and nothing comes to mind. $$\arcsin x + \arccos x = \frac{\pi}{2}$$
Anonymous
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If $20x=\pi$, what is $\frac{\cos 4x - \cos 8x}{\cos 4x\cdot \cos 8x}$?

If $20x=\pi$, what is $$\frac{\cos 4x - \cos 8x}{\cos 4x\cdot \cos 8x}?$$ I've tried using the factor formula on the numerator but I haven't managed to get anywhere with it... This is a multiple choice question with options $4$, $2$, $1$, $-1$, and…
choly
  • 43
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Solve the equation $\cos^n(x) - \sin^n(x)=1$

Solve the equation $\cos^n(x) - \sin^n(x)=1,n \in \mathbb{N}-\{0\}$ If $n$ is even then $\cos^n(x) = \sin^n(x)+1$ is only possible if $\sin(x)=0$ therefore the solution is $x=k\pi, k \in \mathbb{Z}$. I'm having problems with $n$ odd…
user261263
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Inverse trigonometry - how does this work?

I'm learning inverse trigonometry functions from reference books, and certain questions bother me... They're of the type - Prove that $$\arctan(a) + \arctan(b) + \arctan(c) = \pi$$ And they're usually done by adding the angles in terms of…
Shahe Ansar
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Solve sum for theta

Is there any way to solve the following sum of trigonometric functions for theta without using a solver? $$25\sin(\theta)-1.5\cos(\theta)=20$$
mjgpy3
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Polar to cartesian form of $r=\tan(2θ)$

I have attempted converting $r=\tan(2θ)$ to cartesian coordinates: $$r=\frac{2\sin(θ)\cos(θ)}{\cos^2(θ)-\sin^2(θ)}$$ $$r=\frac{2r\sin(θ)r\cos(θ)}{r^2\cos^2(θ)-r^2\sin^2(θ)}$$ $r^2 = x^2 + y^2\\ x = r \cos \theta\\ y = r \sin…
User3910
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Show that $\tan{(\pi/7)} \tan{(2\pi/7)}\tan{(3\pi/7)}=\sqrt{7}$

I tried in this way.$\tan(a+b)=\frac{(\tan a + \tan b)}{1 - \tan a \tan b }$value of $\tan \frac{\pi}{7}$ is coming in decimal.what to do
san092
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Why is the value of $\frac{\sin\theta}{\sin\sin\theta}$ close to the number of degrees in a radian?

Why is $\cfrac{\sin\theta}{\sin\sin\theta}$ very close to 1 radian $(57.2958°)$? I tested this out for $\theta=1^{\circ}$ and $\theta=45^{\circ}$ and various other angles, but I always seem to get about the same answer.
Ju M.
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Solve the equation $\tan(2x) = 1+\tan(x)$

I am trying to solve the equation $$\tan(2x) = 1+\tan(x).$$ I have tried putting $u = tan(x),$ and $tan(2x) = \frac{2u}{1-u^2}$ so that $-u^3 + u^2 + 3u = 1,$ but I can't find any roots that would help me. I have also tried using all the…
t-brum
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Solving $\sin{3x} = \sin{x}$: question about the $k2\pi$ part

Why is $$\sin{3x} = \sin{x}$$ equivalent to $$3x = x + k2\pi$$ $$3x = \pi - x + k2\pi$$ and not $$3x + k2\pi = x + k2\pi$$ $$3x + k2\pi = \pi - x + k2\pi$$
user360150
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How do I determine the point on a square inside a circle depending on an angle?

I can't figure out how to find $(s_x, s_y)$ (see picture, the blue marked intersection). I have $\alpha$, and the square is perfectly inside the square. Assume the radius is 1, since that isn't very important. Hobby programmers aren't the best at…
snowflake
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How to find the period of $\tan2x + \cos2x$?

I need to find the period of the following trigonometric function: $$f(x) = \tan2x + \cos2x$$ Any suggestions?
Kevin
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$\tan^{-1}x$, $\tan^{-1}y$, $\tan^{-1}z$ are in arithmetic progression, as are $x$, $y$, $z$. Show ...

$\tan^{-1}x, \tan^{-1}y, \tan^{-1}z $ are in arithmetic progression, as are $x$, $y$, $z$. (We assume $y \ne 0,1,-1$.) Show: $x$, $y$, $z$ are in geometric progression $x$, $y$, $z$ are in harmonic progression. $x=y=z$ $(x-y)^2 +(y-z)^2+(z-x)^2…
user108430
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Find the value $\tan^{-1}\left(\frac{1}{\sqrt2}\right) - \tan^{-1}\left(\frac{\sqrt{5 - 2{\sqrt6}}}{1+ \sqrt{6}}\right)$

The value of $$\tan^{-1}\left(\frac{1}{\sqrt2}\right) - \tan^{-1}\left(\frac{\sqrt{5 - 2{\sqrt6}}}{1+ \sqrt{6}}\right)$$is equal to $\frac{\pi}{6}$ $\frac{\pi}{4}$ $\frac{\pi}{3}$ $\frac{\pi}{12} $ $$\tan^{-1}\left(\frac{1}{\sqrt2}\right) -…
Aakash Kumar
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