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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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Find $\cos(x-y)$ if $\cos x + \cos y =2$

I had a question in my Math mcq test. If $\cos x + \cos y = 2$ find the value of $\cos(x-y)$. I couldn't get a way to calculate the value. So I just substituted $x = y = 0$. (It seemed obvious to me) So I got $\cos(x-y) = \cos 0 = 1$. But can we…
Sudhanshu
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Trigonometric functions and inverse functions

Can we write $\sin x > a$ as $x > \arcsin a$. Please explain the process. Is it possible for all ratios with any inequality sign.
user168467
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From $\tan(1/A) = \tan(1/B) + \tan(1/C)$ to $A + B + C = ABC$

In this recent question, the equation $$\tan\left(\frac{1}{A}\right) = \tan\left(\frac{1}{B}\right) + \tan\left(\frac{1}{C}\right)$$ is said to imply $$A + B + C = ABC$$ without any stated constraints. Where does this come from, and is it even…
Doubt
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Trignometry-Prove that $(\csc\theta - \sec\theta )(\cot \theta -\tan\theta )=(\csc\theta +\sec\theta )(\sec\theta ·\csc\theta -2)$

Prove that $$(\csc\theta - \sec\theta )(\cot \theta -\tan\theta )=(\csc\theta +\sec\theta )(\sec\theta ·\csc\theta -2)$$ I tried solving the LHS and RHS seperately but they were not coming out to be equal. Please help me answer this question. And…
geek101
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Establishing an identity involving cotangent and cosecant

$$\frac{\csc(x)-1}{\cot(x)}=\frac{\cot(x)}{\csc(x)+1}$$ Once again, "Professor Google" provides an example that's different enough that I can't solve "my" problem. I'm beginning to think that Google does this me on purpose.
Jim Irwin
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Trigonometric Arithmetic Progression

If $a$, $b$, $c$ are in arithmetic progression, prove that $$\cos A \cot\frac{A}{2} \qquad \cos B \cot \frac{B}{2} \qquad \cos C \cot\frac{C}{2}$$ are in arithmetic progression, too. Here, $a$, $b$, $c$ represent the sides of a triangle and $A$,…
Apoorv Jain
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What trig identities must one know to derive the others?

My TA told me in problem section one day that every trig identity could be derived from just 2: the Pythagorean identity and the double-angle identity (or he might have said the half-angle identity). I'm a bit dubious that every trig identity could…
user165538
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How am I supposed to work this out, or do I have to memorize?

When simplifying a trigonometric expression, say, $\sin^2 \theta$ / $\cos^2 \theta$ - I remember that sin over cos is equal to tan. However, what other identities, such as the one mentioned above, do I need to know in general? Is their a way to…
sasha
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How to solve $3 - 2 \cos \theta - 4 \sin \theta - \cos 2\theta + \sin 2\theta = 0$

I have got a bunch of trig equations to solve for tomorrow, and got stuck on this one. Solve for $\theta$: $$3 - 2 \cos \theta - 4 \sin \theta - \cos 2\theta + \sin 2\theta = 0$$ I tried using the addition formula, product-to-sum formula, double…
duci9y
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Prove this trig identity

$$\sin^2(x) - \cos^2(x) - \tan^2(x) = \frac{2\sin^2(x) - 2\sin^4(x) - 1 }{ 1-\sin^2(x)}.$$ I tried this but I can't figure out how they got $-2\sin^4(x)$.
user165018
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Prove trigonometry identity for $\sec x -\sin x$

I'm trying to prove this equality but I' stuck at the second step. Please give me some hints or other ways to proceed. \begin{gather}\frac{\tan^2x + \cos^2x}{\sin x+ \sec x} \equiv \sec x - \sin x \\ \sin x = 0 \\ \cos x = y…
Luther
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What is an algorithm for making text form a circle

Ok it's beyond the scope of this programming exercise, but I want to create a loop that will allow me to input any number of characters and the loop gets each character in the string and places it at regular intervals at specific coordinates in a…
John Conner
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If $\alpha, \beta \in [0,\pi]$ then the minimum value of $\sin(\frac{\alpha +\beta}{2})$ is...

Problem : If $\alpha, \beta \in [0,\pi]$ then the minimum value of $\sin(\frac{\alpha +\beta}{2})$ is a) $\frac{\sin\alpha +\sin\beta}{2}$ b) $|\sin\alpha -\sin\beta|$ c) $\frac{\cos\alpha +\cos\beta}{2}$ d) $|\cos\alpha -\cos\beta|$…
user108258
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Trigonometry - simplifying a given equation

Question: $$\tan 9 - \tan 27 - \tan 63 + \tan 81$$ Answer I'm getting : 0 What I did: Well I clubbed together $\tan 9$ and $\tan 81$ and $\tan 27$ and $\tan 63$ (took out negative as common). Then using the identity for $\tan (A+B)$, I rearranged to…
Gummy bears
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Double Angle Trigonometry Question

So there is this question which consists of 2 parts. $$ a) \text{ Simplify } \frac{\sin2x}{1+\cos2x} \\ b) \text{ Hence, find the exact value of tan 15.} $$ So far I've discovered that $ \text{a)} \tan x $ But I have no idea how to begin on part…
Sam Chahine
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