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 $x$ and $y$

If $\frac{\tan 8°}{1-3\tan^{2}8°}+\frac{3\tan 24°}{1-3\tan^{2}24°}+\frac{9\tan 72°}{1-3\tan^{2}72°}+\frac{27\tan 216°}{1-3\tan^{2}216°}=x\tan 108°+y\tan 8°$, find x and y. I am unable to simplify the first and third terms. I am getting power 4…
user167045
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How to find the exact value of the cosine of 50 degree angle

I want to know the exact value of $\cos 50^\circ$. Actually I have already tried lot of times to solve but I cannot find the exact value of $\cos 50^\circ$.
Peak Mang
  • 43
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Points of intersection of $\sin x$ and $\cos x$

I'm trying to find the points of intersection of $\sin x$ and $\cos x$ between $0$ and $2\pi$. I've tried but I keep getting 4 solutions... Would someone please be able to take me through the process? I squared $\sin x$ and $\cos x$ and then got…
Bob
  • 163
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Find solutions to $\cot(x)+\csc(x)=\sqrt3$ in range $[0,2\pi]$

What is the best way to do the above? Are there any tricks I should be aware of. I know how to simplify it to $\dfrac{\cos(x)}{\sin(x)} + \dfrac{1}{\sin(x)} = \sqrt{3}$ so multiplying both sides by $\sin(x)$, we get $\cos(x)+1=\sqrt{3}\sin(x)$. But…
higgs241
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Hard question in simple trigonometry

This question is from S.L.LONEY- If $\tan(45°+\frac{y}{2})=\tan^3(45°+\frac{x}{2})$, prove that $\frac{\sin y}{\sin x}=\frac{3+\sin^2x}{1+3\sin^2x}$. I don't know what to do. I am getting nasty expressions but nothing in $\sin x$. Thanks.
user167045
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Identities on $\cos n\theta$ and $\sin n\theta$

How to prove that: $$\cos{n\theta}=\cos^n{\theta}- \binom {n} {2}\cos^{n-2} \theta \cdot \sin^2 \theta+ \binom {n} {4}\cos^{n-4} \theta \cdot \sin^{4} \theta -\cdots$$ $$\sin n\theta = \binom {n} {1}\cos^{n-1} \theta \cdot \sin \theta - \binom {n}…
Victor F Salazar G
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How to prove this trigonometry

I need to prove that $$\cos^2(\beta -\gamma)+ \cos^2( \gamma - \alpha) +\cos^2(\alpha -\beta) = 1+2 \cos(\beta- \gamma) \cos( \gamma - \alpha)\cos(\alpha -\beta) $$ To do this I have used the formula $2 cos^2 \theta = 1+ \cos2\theta$ But after…
Numerical Person
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Prove $\sin(45^°) + \sin(15^°) = \sin(75^°)$

I rewrote the statement as $$ \sin(30^° + 15^°) + \sin(15^°) = \cos(15^°). $$ Then I got $$ (\sqrt{3}-2) \sin(15^°) = \cos(15^°). $$
user233631
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Showing the identity: $\tan \alpha + 2 \tan 2\alpha + 4 \tan 4\alpha = \cot \alpha − 8 \cot 8\alpha$

My knowledge of trigonometry is still insufficient to resolve this problem. Any help would be greatly appreciated. $$\tan \alpha + 2 \tan 2\alpha + 4 \tan 4\alpha = \cot \alpha − 8 \cot 8\alpha$$
ignaciotr
  • 51
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Determine height/width of rectangle in perspective

I have the following situation. I've got a 2d plane in which I have drawn a rectangle (red). This is done by picking a point (big red dot), and using the vanishing points calculated by some other lines on the image, and a set width/height of the…
appel
  • 151
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What is $\sin^2(x)$ equal to?

Let's take the sine of $30^\circ$ which is one-half. If you take $\sin^2(30^\circ)$, would that just be the sine of $900$? Or would it be equal to one-quarter, or would it be equal to something completely different?
Billjk
  • 201
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Prove the following trigonometric identity

$$\frac{\tan{(\frac{\pi}{4}+x)}-\tan{(\frac{\pi}{4}-x)}}{\tan{(\frac{\pi}{4}+x)}+\tan{(\frac{\pi}{4}-x)}} = 2\sin{x}\cos{x}$$ ============== On L.H.S, I've tried to write it using the sum and difference formula so it becomes $$\frac{\dfrac{1+\tan…
Titration
  • 33
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Show that the substitution $t=\tan\theta$ transforms the integral ${\int}\frac{d\theta}{9\cos^2\theta+\sin^2\theta}$, into ${\int}\frac{dt}{9+t^2}$

To begin with the $d\theta$ on the top of the fraction threw me off but I'm assuming it's just another way of representing: $${\int}\frac{1}{9\cos^2\theta+\sin^2\theta}\,d\theta$$ I tried working…
Thomas Winkworth
  • 641
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Proving that $\cos(\pi-\phi)=-\cos\phi$ geometrically

I want to geometrically prove that $\cos(\pi-\phi)=-\cos\phi$ without resorting to the unit circle or trigonometric formulas, but have difficulties figuring it out. It's easy enough to do the sine, however: you draw a right triangle to complement…
dpq
  • 141
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$ \sum_{m=1}^{6}\frac{1}{\sin \left\{\theta+\left(m-1\right)\cdot \frac{\pi}{4}\right\} \sin \left\{\theta+m\cdot \frac{\pi}{4}\right\}} = 4\sqrt{2}$

If $\displaystyle 0 < \theta < \frac{\pi}{2}$ and $\displaystyle \sum_{m=1}^{6}\frac{1}{\sin \left\{\theta+\left(m-1\right)\cdot \frac{\pi}{4}\right\}\cdot \sin \left\{\theta+m\cdot \frac{\pi}{4}\right\}} = 4\sqrt{2}$. Then value of…
juantheron
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