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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Trigonometric identities --- working on both sides of the equation at once

When solving trigonometric identities, you aren't allowed to work on both sides of the equation at once. The reason for this is that if you do arrive at a valid conclusion, it doesn't provide the validity of the initial equation - it just proves…
user26649
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Elegant Trigonometric Sums

While studying characters of a finite field and the Polya-Vinogradov inequality, I've found some nice identities (verified by simulations) that I'm not sure how to prove. They seem to be related to Chebyshev's polynomials of the second kind. The…
Ofir
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How prove this $\cot(\pi/15)-4\sin(\pi/15)=\sqrt{15}$

I need some help with this demonstration, please I have tried with some identities but nothing. I wanted to use this $$\sin(\pi/15)\cdot \sin(2\pi/15)\cdots\sin(7\pi/15)=\sqrt{15}$$
Carlos
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Squaring a trigonometric inequality

A very, very basic question. We know $$-1 \leq \cos x \leq 1$$ However, if we square all sides we obtain $$1 \leq \cos^2(x) \leq 1$$ which is only true for some $x$. The result desired is $$0 \leq \cos^2(x) \leq 1$$ Which is quite easily obvious…
user242373
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$\sin(2\arccos(x))$, please help me understand how to do these kind of problems.

We need to be able to transform this equation to get rid of the trig functions. To better explain this, this is how the problem before this one was done. (I checked the answer, i got this one right.) $$\sin(\arccos(x))$$ We substitute $\arccos(x)$…
Steven Rogers
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How prove $\sin \left( \alpha+\frac{\pi }{n} \right) \cdots \sin \left( \alpha+\frac{n\pi }{n} \right) =-\frac{\sin n\alpha}{2^{n-1}}$?

How prove $$\prod_{k=1}^{n}\sin \left( \alpha+\frac{\pi k }{n}\right) =-\frac{\sin n\alpha}{2^{n-1}}$$ for $n \in N$?
piteer
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Expressing $\sin\theta$ and $\cos\theta$ in terms of $\tan(\theta/2)$

This is the question: Let $t = \tan(\theta/2)$. Express the following functions in terms of $t$: $\tan \theta$ $\sin\theta$ $\cos\theta$ I know that for part (1), $$\tan\theta = \frac{2t}{1-t^2}$$ How do I get parts (2) and (3)? If $\tan\theta =…
user163990
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Is $\sin^3 x=\frac{3}{4}\sin x - \frac{1}{4}\sin 3x$?

$$\sin^3 x=\frac{3}{4}\sin x - \frac{1}{4}\sin 3x$$ Is there any formula that tells this or why is it like that?
Andrew
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How to get sine / cosine value out of tangens

I know that: $\tan(\alpha) = 1/2$. How can I get clean values for sine / cosine without the calculator? Is there a relationship? I know that $\sin(\arctan(1/2))$ is a way ... But I hope you get the point. Thank you!
DAS
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Evaluate trig functions without a calculator

My precalculus test asked me to determine which was greater: $\tan (53)$ or $\sec (38)$. I looked at it like this, but it seems so close that it's difficult to imagine that they would ask this: $\tan (45)$ is 1 and $\tan (60)$ is $\sqrt{3}$, so…
user163862
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A nice trignometric identity

How to prove that: $$\cos\dfrac{2\pi}{13}+\cos\dfrac{6\pi}{13}+\cos\dfrac{8\pi}{13}=\dfrac{\sqrt{13}-1}{4} $$ I have a solution but its quite lengthy, I would like to see some elegant solutions. Thanks!
Mathslover
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Do you ever say that the amplitude is negative?

If you have a trig function $f(x) =- 3\sin (2x) + 1$ then would you ever say that the amplitude is negative? I've seen it stated that it can be negative or that amplitude is a distance so it should only be a positive value. I'm wondering if it…
tazboy
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What does $\sin(\sin(x))$ mean?

What does an equation like $\sin(\sin(x))$ mean? I know it can be seen as a composite function $f(f(x))$, where $f(x)=\sin(x)$. Is there a way to simplify functions like this, and where will this be used? Thanks in advance. P.S. I have looked at the…
Anonymous Computer
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Proving a trig identity: $\frac{\sin(A + B)}{\sin(A - B)}=\frac{\tan A + \tan B}{\tan A - \tan B}$

I'm learning about trig identities, and I'm struggling to prove that two expressions are equal: $$ \frac{\sin(A + B)}{\sin(A - B)}=\frac{\tan A + \tan B}{\tan A - \tan B} $$ How do you go about proving this? I know about compound angles - i.e. the…
hohner
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Solving $\cos x=x$

I would like to know how can we solve the equation $\cos x = x$, without graphing. I know that there would only be one solution, that is obvious, that too in between $0$ and $\frac{\pi}{2}$. Is there any real expression in finite terms [perhaps we…
Sawarnik
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