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.

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Solve $\cos2A+\sin A=2$

So they tell me solve $\cos2A+\sin A=2$. Ok guys I found the problem :/ I was wondering why he gave us this question but its actually $$\cos2A+\sin A=-2$$ and I simplified to $-2x^2+x+3=0$, which has solutions $x= 3/2$, $x=-1$ So I'm guessing then…
user66288
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Elementary proof of Bhaskara I's approximation: $\sin\theta=\frac{4\theta(180-\theta)}{40500-\theta(180-\theta)}$

This site has seen umpteen questions about efficient ways to calculate the sine of an angle. But a remarkable formula was given in ~CE 615 by the Indian mathematician Bhaskara I in his Mahabhaskariya. The formula goes as…
Sandhi
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Wolfram Alpha's Mysterious Trig Abilities...

This is a very straight-forward question: I'm trying to simplify $$\sin(2\pi t +\pi/4) + \sin(2\pi t -\pi/4)$$ and failing at it: $\sin(2\pi t +\pi/4) + \sin(2\pi t -\pi/4)$ $2\sin(2\pi t)\cos(\pi/2)$ by sum $\to$ product = ZERO Wolfram alpha…
Griffin
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Solving a system of equations with trig

How do you solve a system of the following form: $$a = 2\sin(x) - \sin(y) + \sin(x+y)\tag1$$ $$b = 2\sin(y) - \sin(x) + \sin(x+y)\tag2$$ where $a,b$ are constants, and $x,y$ the variables I'd like to solve for. Subtracting $(1)-(2)$ gives an…
Willywonka
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Show that $\cos^220^\circ-\cos20^\circ\sin10^\circ+\sin^210^\circ=\frac34$

The original exercise is to Prove that $$4(\cos^320^\circ+\sin^310^\circ)=3(\cos20^\circ+\sin10^\circ)$$ Dividing both sides by $\cos20^\circ+\sin10^\circ$ leads me to the problem in the question title. I've tried rewriting the left side in terms…
user170231
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value of $x$ in Trigonometric equation

Find real $x(0
jacky
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Different solutions of trigonometric equations

Please take a look at this trigonometric equation, $\cos9x\cos7x = \cos5x\cos3x$ To solve this equation, we can proceed as, $2\cos9x\cos7x = 2\cos5x\cos3x$ or, $\cos(9x+7x)+\cos(9x-7x) = \cos(5x+3x)+\cos(5x-3x)$ or, $\cos16x+\cos2x =…
Masroor
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Different answers with $\sec(x) = 2\csc(x)$

My son and I were solving this last night and we get different answers depending on which identities we use. The question also did specify $0 \leqslant x < 2\pi$ Here's our work: $$\sec x = 2 \csc x$$ $$\frac 1 {\cos x} = \frac 2 {\sin x}$$ cross…
rrauenza
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How to find the exact value of $ \cos(36^\circ) $?

The problem reads as follows: Noting that $t=\frac{\pi}{5}$ satisfies $3t=\pi-2t$, find the exact value of $$\cos(36^\circ)$$ it says that you may find useful the following identities: $$\cos^2 t+\sin^2 t = 1,\\ \sin 2t = 2\sin t\cos t,\\ \sin…
Maximilian1988
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Further simplify $\tan(\alpha+\beta)-\tan(\beta)$

I came up with the formula \begin{align*} \tan(\alpha+\beta)-\tan(\beta) \end{align*} but I keep wondering, whether it's possible to further simplify this, into for example only using the $\tan$ once. I tried using the addition theorems for…
Ian H.
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what does "arc" in arcsin, arccos, arctan stands for

I was just wondering, what does the "arc" in arcsin, arccos, arctan stands for? Is there any particular reason why it is named the way it is?
Thor
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Is there a simpler way to determine m, n, p, such that the following holds for all reals?

I am given the following equation and I am asked to find $m$, $n$ and $p$, so that the equation holds for all reals: $$ \sin^4x + \cos^4x + m(\sin^6x + \cos^6x) + n(\sin^8x + \cos^8x) + p(\sin^{10}x + \cos^{10}x) = 1, \space \forall x \in \mathbb R…
cheez3d
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Prove this $\sin(nx)$ identity

Without induction How to prove that $$\sin(2nx)=2n\sin x \cos x \prod_{k=1}^{n-1}\left(1-\frac{\sin^2(x)}{\sin^2\frac{k\pi}{2n}}\right)$$ for Natural n I Tried by several way and the last try is to use euler formula which lead me…
mnsh
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How do I find out the value of $p$ for $\sin p + \cos p = 0$?

I tried doing this by putting $$\frac{ \sin p}{\sqrt2} +\frac{ \cos p}{\sqrt2} = 0 $$ which implies $$ \sin(p + π/4) = \sin 0 $$ which implies $p+ π/4 = nπ $. Now according to the question I'm solving $p = 2πt/T$. I need to get the relation $t =…
Hema
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How can I to solve $ \cos x = 2x$?

I would like to get an approx. solution to the equation: $ \cos x = 2x$, I don't need an exact solution just some approx. And I need a solution using elementary mathematics (without derivatives etc).
gen
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