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 - Double angle

I have tried to solve this problem, but everything that I try does not work. Please help me solve this equation: $$\cos {6}x + 2 = 5\sin {3}x$$ Thanks :)
LiziPizi
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Simplify a trigonometric expression

I've been simplying a slew of trigonometric expressions and most of them fall out pretty clearly, but this one has been giving me fits: $$\frac{(\sec x - \tan x)^{2} + 1}{\sec x\csc x - \tan x\csc x}$$ Here's my best attempt so…
Tom M
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Should cosecant be defined as $\csc \theta = \frac{1}{\sin \theta}$, specifying the constraint: $\sin \theta \neq 0$?

I'm studying trigonometry on my own, and I keep noticing that the trigonometric functions are never defined with constraints to deal with divide-by-zero issues. As an example, I've seen cosecant defined like this: $\csc \theta = \frac{1}{\sin…
John H
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Solving $\cos^2{\theta}-\sin{\theta} = 1$

Can someone please help me solve this? $$\cos^{2}{\theta}-\sin{\theta} = 1, \quad\theta\in[0^{\circ}, 360^{\circ}]$$
barryedx
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Solving $a \sin 2x = \sin (x + \gamma)$

I am trying to solve the following equation: $$a \sin 2x = \sin (x + \gamma)$$ or, equivalently: $$2 a = \frac{\cos \gamma}{\cos x} + \frac{\sin \gamma}{\sin x}$$ where $a$ and $\gamma$ are constants. I tried for a long time, and searched the web a…
fishinear
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If $a_1+a_2\sin x+a_3\cos x+a_4\sin 2x+a_5\cos 2x=0$ is an identity in $x$,then prove that $(a_1,a_2,a_3,a_4,a_5)=(0,0,0,0,0)$

If $a_1+a_2\sin x+a_3\cos x+a_4\sin 2x+a_5\cos 2x=0$ is an identity in $x$,then prove that $(a_1,a_2,a_3,a_4,a_5)=(0,0,0,0,0)$ I tried:$a_1+a_2\sin x+a_3\cos x+a_4\sin 2x+a_5\cos 2x=0$ $a_1+a_2\sin x+a_3\cos x+2a_4\sin x\cos…
Brahmagupta
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Given $x\in \left(0; \frac\pi2\right)$. Prove that $\sin x>\frac{2x}{\pi}$

Given $x\in \left(0; \frac\pi2\right)$. Prove that $$\sin x>\frac{2x}{\pi}$$ This is my try: Let $y=\sin x-\frac{2x}\pi\implies y ' = \cos x - \frac2\pi\implies y ''=-\sin x <0; \forall x\in \left(0; \frac\pi2\right)$ $\implies…
mja
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How would I find a point on a sphere with a UV coordinate?

I'd like to do the opposite of the example specified here: https://en.wikipedia.org/wiki/UV_mapping Can somebody explain to me how to do it? Thanks, For any point $P$ on the sphere, calculate $\hat d$, that being the unit vector from $P$ to the…
Bushes
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Equation of the form $\tan(\alpha)=\cos(\alpha+C)$ where $C\in\mathbb{R}$

I have seen the following math problem posed online by a high school student (knowing their material, most likely it wasn't given as an exercise): Find the solutions for the equation $$\tan(\alpha)=\cos(\alpha+33.44^{\circ}).$$ Oddly, the solution…
DailyRayleigh
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Solve trigonometric inequality $ \sin x \sin 2x - \cos x \cos 2x > \sin 6x $

Solve this trigonometric inequality: $$ \sin x \sin 2x - \cos x \cos 2x > \sin 6x $$ My steps: $$ \cos x \cos 2x - \sin x \sin 2x < - \sin 6x $$ $$ \cos 3x < \sin (-6x)$$ $$ \cos 3x < \cos (\frac{\pi}{2}+6x) $$ From this we get: $$ 3x >…
Gjekaks
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Proving an identity using reciprocal, quotient, or Pythagorean identities.

I've been trying to prove this for a while, to no avail. I am only allowed to use pythagorean, quotient, and reciprocal identities: $$\frac{\tan \theta}{1 + \cos \theta} = \sec \theta \csc\theta(1-\cos \theta)$$ I've tried converting $\tan \theta$…
DMan
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Solve trigonometric equation $ 3 \cos x + 2\sin x=1 $

Solve trigonometric equation: $$ 3 \cos x + 2\sin x=1 $$ I tried to substitue $\cos x = \dfrac{1-t^2}{1+t^2}, \sin x = \dfrac{2t}{1+t^2}$. Yet with no results.
Gjekaks
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Can height of a curb be determined by the angles of scratches on the perimeter of a wheel that struck a curb?

With any level of certainty, can the angles of scratches on the outermost edge of a wheel of known diameter be used to calculate the height of a curb, which the wheel struck at low velocity? The wheel didn't strike the curb head on, but rather…
Michael Prescott
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Solve this trigonometric equation $ \sin2x-\sqrt3\cos2x=2$

Solve equation: $$ \sin2x-\sqrt3\cos2x=2$$ I tried dividing both sides with $\cos2x$ but then I win $\frac{2}{\cos2x}$.
Gjekaks
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If $ \arcsin x+ \arcsin y=\frac{\pi}{2}$, prove that $ x^2+y^2=1$.

If $ \arcsin x+ \arcsin y=\frac{\pi}{2}$, prove that $$ x^2+y^2=1$$ I tried taking sine of both the sides, I only come to this result: $$x^2 + y^2 -2x^2y^2 + 2xy\sqrt{(1-y^2)(1+x^2)}=1.$$
Gjekaks
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