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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Trigonometry + Geometry

In the given triangle we have this point $O$ such that $\angle OAB=\angle OBC=\angle OCA=\omega$ Hence prove that $\cot\omega=\cot A+\cot B+\cot C$. I figured out the RHS by using sine and cosine identities but the LHS couldn't be worked out by me.…
Harsh Sharma
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Can you solve a trig equation with a variable both inside a trig function and outside one?

I have the equation: $$d=\frac{t}{2}-\frac{sin(t)}{4}$$ I'm completely failing at how to get this in terms of $t$ I only care about it for values of $0
Caps
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In any triangle, $a^2(B-C)+b^3\cos (C-A)+c^2\cos (A-B)=3abc$.

Let $A,B,C$ by the angles of a triangle, and let $a,b,c$ be the length of the corresponding opposite sides. How can you prove that $$a^3\cos(B-C) + b^3\cos(C-A) +c^3\cos(A-B) = 3abc?$$ I divided both sides by $abc$ and then tried to open the cosine…
Harsh Sharma
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How many solutions has the equation $\sin x= \frac{x}{100}$ ?

How many solutions has the equation $\sin x= \frac{x}{100}$ ? Usually when I was asked to solve this type of problem, I would solve it graphically but this one seems to be trickier. It doesn't seem wise to put $f(x)=\sin x$ and $g(x)=\frac{x}{100}$…
lmc
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Reducing $\tan\frac{\pi}{16} + 2\tan\frac{\pi}{8} +4$ to $\cot\frac{\pi}{16}$

$$\text{The value of}\quad\tan\frac{\pi}{16} + 2\tan\frac{\pi}{8} +4 \quad\text{is equal to _______.}$$ (Answer: $\cot\frac{\pi}{16}$) I solved the question by the identity $$\tan \phi = \cot\phi-2\cot 2\phi$$ and got the right answer.…
Harsh Sharma
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If $ax+b\sec(\tan^{-1}x)=c$ and $ay+b\sec(\tan^{-1}y)=c$, then ...

After having struggled yesterday with this as much as I could, I am posting this problem here: If $ax+b\sec(\tan^{-1}x)=c$ and $ay+b\sec(\tan^{-1}y)=c$, then prove that $$\frac{x+y}{1-xy}=\frac{2ac}{a^2-c^2}$$ My attempt: Comparing both the…
GRrocks
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How to solve trig equations and get all the solutions using graphs, $\cos(2x-\pi/3)=\cos(x)$

The question is to solve $$\cos\left(2x-\frac{\pi}{3}\right)=\cos(x)$$ I originally approached this using the addition formulae but the mark scheme showed a way by first replacing $x$ on the right with $2\pi-x$ and I understand this is due to the…
Radhika
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Solving for $x$ in $\tan(3x) \tan (2x)= 1$

If $$\tan(3x) \tan(2x)= 1$$ Then $x$ is equal to Attempt: I used the '$\tan$' identity but it showed no results. The identity: $$\frac{\tan(2x)+\tan(3x)}{1-\tan(2x)\tan(3x)}$$
Iqbal Jurist
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Proving equation at zero?

I have an equation $$x = \csc(\theta) - \cot(\theta).$$ As $\theta$ approaches zero, $x$ approaches zero. However, trying to solve the equation at zero yields an undefined result. How do I rewrite the equation to be continuous at 0?
MerickOWA
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Prove $\cos^2 x \,\sin^3 x=\frac{1}{16}(2 \sin x + \sin 3x - \sin 5x)$

How would I prove the following? $$\cos^2 x \,\sin^3 x=\frac{1}{16}(2 \sin x + \sin 3x - \sin 5x)$$ I do not know how to do do the problem I do know $\sin(3x)$ can be $\sin(2x+x)$ and such yet I am not sure how to commence.
Fernando Martinez
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The Inverse Trigonometric Functions

I know that if $y=\sin x$ then $\arcsin y=x$; that is, $\arcsin$ is used for the inverse and $\arcsin$ is not a function if we don't restrict the domain of $y=\sin x$ but I don"t get that what is the meaning of arc in "$\arcsin$" because $\arcsin$…
Waqar
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Any more of this $a\arctan\left(\frac{1}{\phi^n}\right)+b\arctan\left(\frac{1}{\phi^m}\right)=\frac{\pi}{4}$ type?

$$\arctan\left(\frac{1}{\phi}\right)+\arctan\left(\frac{1}{\phi^3}\right)=\frac{\pi}{4}$$ $$2\arctan\left(\frac{1}{\phi^2}\right)+\arctan\left(\frac{1}{\phi^6}\right)=\frac{\pi}{4}$$ $$3\arctan\left(\frac{1}{\phi^3}\right)+\arctan\left(\frac{1}{\phi^…
user339807
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Why $\frac{\pi}{12}$ equals to $\frac{\pi}{3} - \frac{\pi}{4}$

I'm going back to basic trigo for the sake of being able to help my kids and also being bad younger at it, I want to be able to overcome that lack of understanding and honestly, I hate unfinish business. So please bear with me if you feel my…
Andy K
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Prove that $\prod^{n}_{k=1}\sin\left(\frac{k\pi}{2n+1}\right) = \frac{\sqrt{2n+1}}{2^{n}}$

Prove that $$\prod^{n}_{k=1}\sin\left(\frac{k\pi}{2n+1}\right) = \frac{\sqrt{2n+1}}{2^{n}}$$ $\bf{My\; Try::}$ Let $\displaystyle \cos \left(\frac{k\pi}{2n+1}\right)+i\sin \left(\frac{k\pi}{2n+1}\right)=e^{\frac{ik\pi}{2n+1}}=e^{ik \theta}\;,$…
juantheron
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Show that $\tan x +\frac{1}{2}\tan \frac{x}{2}+\dots + \frac{1}{2^n}\tan \frac{x}{2^n} = \frac{1}{2^n}\tan \frac{x}{2^n}-2\cot(2x) $

Show that $$\tan x +\frac{1}{2}\tan \frac{x}{2}+\dots + \frac{1}{2^n}\tan \frac{x}{2^n} = \frac{1}{2^n}\tan \frac{x}{2^n}-2\cot(2x), \quad (n=0,1,\ldots). $$ I tried to prove this using induction but with no result. I'm not sure how to begin to…
Gjekaks
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