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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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Show that $\cot(\pi/14)-4\sin(\pi/7)=\sqrt7$

Show that $\cot(\pi/14)-4\sin(\pi/7)=\sqrt7$. This problem is from G.M. 10/2016 and I can't solve it. I tried with an isosceles triangle with angles $3\pi/7, 3\pi/7$ and $\pi/7$ and I tried to find a relation between the sides of the triangle but I…
razvanelda
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A "simple" trigonometric computation...

Show that $$\left(1+4\sin^2\frac{\pi}{18}\right) \left(1+4\sin^2\frac{\pi}{6}\right) \left(1+4\sin^2\frac{5\pi}{18}\right)\left(1+4\sin^2\frac{7\pi}{18}\right)=34.$$ Any ideas about how to tackle this?
user368484
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Prove that :$\tan 70 - \tan 20 = 2\tan 40 + 4\tan 10$

Prove that :$\tan 70 - \tan 20 = 2\tan 40 + 4\tan 10$ My Attempt, $$70-20=40+10$$ $$\tan (70-20)=\tan (40+10)$$ $$\dfrac {\tan 70 - \tan 20}{1+\tan 70. \tan 20 }=\dfrac {\tan 40 + \tan 10 }{1-\tan 40. \tan 10 }$$ How should I move on? Please…
Aryabhatta
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Proving trigonometric Identity: $\frac{1+\sin x}{\cos x} = \frac{1+\sin x+\cos x}{1-\sin x+\cos x}$

I would like to try and prove $$\frac{1+\sin x}{\cos x} = \frac{1+\sin x+\cos x}{1-\sin x+\cos x}$$ using $LHS=RHS$ methods, i.e. pick a side and rewrite it to make it identical to the other side. I found a quick way by doing this: $$LHS =…
Argon
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How would a triangle for sin 90 degree look

I am studying trigonometry in my school and learned that in a triangle the side opposite to angle theta should be taken as perpendicular side - hypotenuse remains the same and the third remaining side is the base. The same is required for…
Programmer
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Intersection with object on ellipsoid path

In my game I need to calculate the time when a ship arrives at the planet. What is known is ship's starting position and velocity (which is constant), and planet’s position at a given time (it follows an elliptic path). To be more…
Betalord
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Why do we omit the negative sign when finding basic angle when solving trigonometric equations?

Suppose I'm asked to solve for $\cos \theta = -0.5$, for $\theta$ between $0^\circ$ and $360^\circ$ inclusive. I am told that the first step would be to buy the basic angle, $\alpha$. The way I am told to do this is by omitting the negative sign…
Charlz97
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Singapore math olympiad Trigonometry question: If $\sqrt{9-8\sin 50^\circ} = a+b\csc 50^\circ$, then $ab=$?

$$\text{If}\; \sqrt{9-8\sin 50^\circ} = a+b\csc 50^\circ\text{, then}\; ab=\text{?}$$ $\bf{My\; Try::}$ We can write above question as $$\sin 50^\circ\sqrt{9-8\sin 50^\circ} = a\sin 50^\circ+b$$ Now for Left side, $$\sin 50^\circ\sqrt{9-8\sin…
juantheron
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How to solve $x+\sin(x)=b$

How can I solve $x+\sin(x)=b$ for $x \in [0,π]$? We take $b \in [0,π]$. I don't know how to find the solution.
Babyblog
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Prove that $ \frac{1}{2}\cot^{-1}\frac{2\sqrt[3]{4}+1}{\sqrt{3}}+\frac{1}{3}\tan^{-1}\frac{\sqrt[3]{4}+1}{\sqrt{3}}=\dfrac{\pi}{6}. $

Prove that $$ \frac{1}{2}\cot^{-1}\frac{2\sqrt[3]{4}+1}{\sqrt{3}}+\frac{1}{3}\tan^{-1}\frac{\sqrt[3]{4}+1}{\sqrt{3}}=\dfrac{\pi}{6}. $$
Mathephobia
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Evaluate $\sum_ {n=1}^{\infty} \cot^{-1}(2n^2)$

I was trying to solve up this equation but couldn't move ahead. $$\sum_ {n=1}^{\infty} \cot^{-1}(2n^2)$$ I wrote the expression as $$\sum_ {n=1}^{\infty} \tan^{-1}\left( \frac{1}{2n^2}\right)$$ I wanted to change the expression into such a form such…
Harsh Sharma
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If $\sin x + \csc x =2 \tan x$. Find value of $\cos^9x +\cot^9x +\sin^7x$

Problem: If $\sin x+\csc x=2\tan x$, Find value of $\cos^9x+\cot^9x+\sin^7x$ Solution: \begin{align*}&\sin x+\csc x=2\tan x \\ &\sin x+\frac{1}{\sin x}=2\frac{\sin x}{\cos x} \\ &\sin^2x+1=2\frac{\sin^2x}{\cos x} \\ &\sin^2x\cos x+\cos…
rst
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Expressing $\sin^4x-\sin^6x$ in another way

I am slightly confused on how one would subtract $\sin^4x-\sin^6x$. I know that $\sin^2x=(1/2)(1-\cos2x)$, so $\sin^4x$ would logically be $[(1/2)(1-\cos2x)]^2=(1/4)(1-2\cos(2x)+\cos^2(2x)$ However the value of $\sin^6x$ eludes me. Would it be…
Fernando Martinez
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Why am I getting two different answers (and the textbook a third) on this 3D trig problem?

Simone is facing north and facing the entrance to a tunnel through a mountain. She notices that a $1515$ m high mountain is at a bearing of $270^\circ$ from where she is standing and its peak has an angle of elevation of $35^\circ$. When she…
user262291
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basics of Trigonometry

I have learned trigonometry, in school but never understood clearly what it is..just mugged up formula's and theorems to get clear the exams. But now i want to know what exactly, is trigonometry from the basics so that i can understand.
Harish Kurup
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