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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How can tan(30°) be irrational and at the same time be a ratio within a triangle?

Dumb question... if tan(30°) is irrational (what I believe it is, it should be $\sqrt 3$), how can it be that, at the same time, is is describing the ratio of two sides within a triangle? The triangle I have in mind looks like this (sorry for the…
volzotan
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How prove $\cos(\frac{2\pi}{17}) + \cos(\frac{18\pi}{17})+\cos(\frac{26\pi}{17})+\cos(\frac{30\pi}{17}) = \frac{\sqrt{17}-1}{4}$

Prove that $\cos(\frac{2\pi}{17}) + \cos(\frac{18\pi}{17})+\cos(\frac{26\pi}{17})+\cos(\frac{30\pi}{17}) = \frac{\sqrt{17}-1}{4}$ Regards that value of $\cos(2\pi/17)$, I can't find the easy way to solve that expression. Even if I had time, I…
miguel747
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Evaluating $\cos^{-1}(\sin(-17))$

I had this question on my test I took today, and I'm confused if my answer's right. I had to find the value of $$\cos^{-1}(\sin(-17))$$ Okay, first, I drew a triangle. And, after, I let a and b for each line related to sin -17, so that we can say…
user547075
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Solving for $k$ in $\cos \frac{\pi}k - \cos \frac{2\pi}k = P$

How do I solve trigonometric equations of the type $$\cos \frac{\pi}k - \cos \frac{2\pi}k = P$$ where P is a real constant within the range of the left side. (By solving, I mean finding the value of k) I could convert this into product of two…
SmarthBansal
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How can I find the hypotenuse and opposite sides of triangle?

Based on the given information how can I find the pixel values of the hypotenuse and opposite sides of the triangle?
pizzarob
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Evaluating $\prod^{100}_{k=1}\left[1+2\cos \frac{2\pi \cdot 3^k}{3^{100}+1}\right]$

Evaluate$$\prod^{100}_{k=1}\left[1+2\cos \frac{2\pi \cdot 3^k}{3^{100}+1}\right]$$ My attempt: $$1+2\cos 2\theta= 1+2(1-2\sin^2\theta)=3-4\sin^2\theta$$ $$=\frac{3\sin \theta-4\sin^3\theta}{\sin \theta}=\frac{\sin 3\theta}{\sin \theta}$$ I did…
jacky
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Trigonometric series sum involving tangents

$$\frac{1}{4}\tan\bigg(\frac{\pi}{8}\bigg)+\frac{1}{8}\tan\bigg(\frac{\pi}{16}\bigg)+\frac{1}{16}\tan\bigg(\frac{\pi}{32}\bigg)+\cdots\cdots \infty$$ Try: $$\cos x\cdot \cos(x/2)\cdot\cos(x/2^2)\cdots\cdots…
DXT
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Simple Trig Problem

I'm a bit stuck on a homework question that I've been assigned. The question is as follows: You are paddling a canoe at a speed of $4$ $km/h$ directly across a river that flows at $3$ $km/h$. $(a)$ What is your resultant speed relative to the…
drokkin
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Solve the equation $\cos^2x+\cos^22x+\cos^23x=1$

Solve the equation: $$\cos^2x+\cos^22x+\cos^23x=1$$ IMO 1962/4 My first attempt in solving the problem is to simplify the equation and express all terms in terms of $\cos x$. Even without an extensive knowledge about trigonometric identities,…
John Glenn
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How to express $\theta$ in terms of $x$ where $3\sin(3\theta+x)=\frac{2.5}{\sin\theta}$?

I tried to solve it by using compound angle formulas but in the end I could not leave $\theta$ alone. It goes like this: \begin{align} & \frac{2.5}{3}=\sin(3\theta+x)\sin\theta \\[10pt] &…
Sayra
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Is the equation $A = \frac{d^2}{8}\left(\alpha-\sin\alpha\right)$ solvable for $\alpha$?

I have been given this equation. $A$ and $d$ are known, and I want to solve to $\alpha$. $$A=\frac{\pi d^2}{4}\cdot\frac{\alpha}{2…
Daniel
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Trig question, can this be solved?

I was wondering is it possible to solve this without assuming that CAD=DAB. As I use the law of sines, trigonometry and have tried to apply law of cosines. However, I cannot see how you can solve this with just using a and $\alpha$.
simplicity
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Trigonometry Exact Value using Half Angle Identity

I have a quick question regarding a little issue. So I'm given a problem that says "$\tan \left(\frac{9\pi}{8}\right)$" and I'm supposed to find the exact value using half angle identities. I know what these identities are $\sin, \cos, \tan$. So, I…
programstudent22
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Can these trig functions be reduced to the pythagorean theorem?

Given a right triangle, and its two legs $a$ and $b$, I know of two ways to find the hypotenuse $c$: Equation 1: $$c=\sqrt{a^2+b^2}$$ Equation 2: $$c=\frac{b}{\sin{(\tan^{-1}{(\frac{b}{a})})}}$$ The first equation is just the pythagorean theorem.…
FuzzyCat444
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Cotangent domain error?

Some time ago I was wondering about the definition of cotangent; namely, that it is both defined as $\frac{\cos x}{\sin x}$ and as $\frac{1}{\tan x}$. However, $\cot(90°)$ and $\cot(270°)$ are equal to 0. Through the first definition I listed of…
Gil Keidar
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