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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Showing Trigonometric Identity

Prove that: $$\cos^2\theta\sin^4\theta=\frac{1}{32}(\cos6\theta-\cos2\theta+2-2\cos4\theta)$$ Attempt: \begin{align*} L.H.S & = \cos^2\theta\sin^4\theta\\ & = \cos^2\theta\sin^2\theta\sin^2\theta\\ & =…
Iaamuser user
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What is meant by a 'pure' wave?

What is meant by a 'pure' wave? I know it might sound like a basic question, but I've never been taught this. I saw that a sine wave is a pure wave. I tried Googling what a pure wave is, but all I get is links regarding Pure Wave Inverters for…
Max Echendu
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Find the minimum value of $\sin^{2} \theta +\cos^{2} \theta+\sec^{2} \theta+\csc^{2} \theta+\tan^{2} \theta+\cot^{2} \theta$

Find the minimum value of $\sin^{2} \theta +\cos^{2} \theta+\sec^{2} \theta+\csc^{2} \theta+\tan^{2} \theta+\cot^{2} \theta$ $a.)\ 1 \ \ \ \ \ \ \ \ \ \ \ \ b.)\ 3 \\ c.)\ 5 \ \ \ \ \ \ \ \ \ \ \ \ d.)\ 7 $ $\sin^{2} \theta +\cos^{2}…
R K
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Solve $\sin(3x)=\cos(2x)$

Question: Solve $\sin(3x)=\cos(2x)$ for $0≤x≤2\pi$. My knowledge on the subject; I know the general identities, compound angle formulas and double angle formulas so I can only apply those. With that in mind \begin{align} \cos(2x)=&~…
bigfocalchord
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Simplify $\frac{4\sqrt{7}}{3}\cos{\left(\frac{1}{3}\arccos{\frac{1}{\sqrt{28}}}\right)}+\frac{1}{3}$

If $\dfrac{2\sqrt{19}}{3}\cos{\left(\dfrac{1}{3}\arccos{\dfrac{7}{\sqrt{76}}}\right)}-\dfrac{1}{3}$ can be simpified to $2\left(\cos{\dfrac{4\pi}{19}}+\cos{\dfrac{6\pi}{19}}+\cos{\dfrac{10\pi}{19}}\right)$. How to simplify …
Bless
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Can I define cotangent as $\cot x=\cos x/\sin x$?

Since $\tan x=\sin x/\cos x$ and $\cot x=1/\tan x$, can we redefine cotangent as $\cot x =\cos x/\sin x$? if we use this definition, we can find this value $\cot \pi/2= 0$. What are the advantages of the second definition?
user42912
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How to express $\tan kx$ as a function of $\tan x$?

I had a problem to express $\tan kx$ as a function of $\tan x$. For example, $\tan 3x=(3 \tan x−\tan^ 3x)/(1−3\tan^2x)$. But in general case, how can I express for example $\tan 10x$ as a powers of $\tan x$? I saw a Chebyshev method from…
curious
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Solve $\frac{\cos x}{1+\sin x} + \frac{1+\sin x}{\cos x} = 2$

I am fairly good at solving trig equations yet this one equation has me stumped. I've been trying very hard but was unable to solve it. Can anyone help please? Thank you. $$\frac{\cos x}{1+\sin x} + \frac{1+\sin x}{\cos x} = 2$$ solve for $x$ in the…
Outlier
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Length of hypotenuse v/s change in height of the opposite

I have always struggled to understand mathematical concepts, and have a very different way of thinking about problems. I suspect this is a very simple problem, but its confusing me a great deal. I made a right triangle where the "opposite" side was…
Cggart
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The maximum and minimum values of the expression

Here is the question:find the difference between maximum and minimum values of $u^2$ where $$u=\sqrt{a^2\cos^2x+b^2\sin^2x} + \sqrt{a^2\sin^2x+b^2\cos^2x}$$ My try:I have just normally squared the expression and got $u^2=a^2\cos^2x+b^2\sin^2x +…
Kartik Watwani
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Problem with sine in a right triangle

Given a triangle $ABC$ with angles a,b & c, prove that if $\sin^2(a) + \sin^2(b) + \sin^2(c) = 2$ then the triangle is right angled (has an angle of $90^o$). If I assume the triangle is right angled and have AB, AC and BC as sides, with BC being the…
MikhaelM
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Is there a way to write this expression differently: $\arctan 1/(1+n+n^2)$?

$$ \arctan\left(\frac{1}{1+n+n^2}\right)$$ My professor wrote this as $$\arctan(n+1) - \arctan(n)$$ I don't understand how this expression is right?
Rocky G.
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Find the value of : $\cos x \cos 2x...\cos 999x$ given that $x=\frac {2\pi}{1999}$

Given $x=\frac {2\pi}{1999}$ Find the value of $$\cos x \cos 2x \cos 3x ...\cos 999x$$ So I tried expanding $\sin {2000x}=2\sin 1000x \cos 1000x$ Then rewriting $\cos 1000x= \cos {(999x+x)}$ No luck so far.
Shubham
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Evaluate $\sin(\frac{\pi}{8})$ and $\cos(\frac{\pi}{8})$

Evaluate $\sin(\frac{\pi}{8})$ and $\cos(\frac{\pi}{8})$ I was just wondering what I am doing wrong, as I don't seem to be arriving at the correct answer for $\sin(\frac{\pi}{8})$ What I did: Let $\theta = \frac{\pi}{8}$ $\cos(2\theta) =…
stariz77
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Prove that $\displaystyle{\frac{\cos A+\cos B - \cos C}{\sin A+\sin B - \sin C}} \geq -\frac{\sqrt{3}}{3}$

All the angles in a triangle $A,B,$ and $C$ are less than $120^{o}$ Prove that $\displaystyle{\frac{\cos A+\cos B - \cos C}{\sin A+\sin B - \sin C}} \geq -\frac{\sqrt{3}}{3}$
Kirthi Raman
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