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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.

29665 questions
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Find the least positive angle satisfying the trigonometric equation

$\sin^3 x+\sin^3 2x+\sin^3 3x=(\sin x+\sin 2x+\sin 3x)^3$. I did solve the question, but my method is highly tedious. I combined the sin and then opened the cubic.... Is there some trick? Something I am missing? Thanks.
user167045
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Substituting a value of sine function in a trigonometric equation

I am trying to really understand trigonometric equations and I've stumbled upon a rather confusing example. Solve the following equation: $\sin x= 2|\sin x|+ {\sqrt{3}}\cos x$ First step is to define the absolute $\sin x$: $$|\sin x| =…
0lt
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Find $a$, when $\tan a$ is given in terms of $\tan1^{\circ}$ and $\tan2^{\circ}$.

If $\tan\alpha = {(1+\tan1°)(1+\tan2°)-2 \over (1-\tan1°)(1-\tan2°) - 2}$ and $\alpha \in (0°, 90°)$ then $\alpha$ is equal to? This is task from my faculty entrance exam workbook. This is mostly high school level and I can only assume that I need…
GreatDuke
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High School Trigonometry ( Law of cosine and sine)

I am preparing for faculty entrance exam and this was the question for which I couldn't find the way to solve (answer is 0). I guess they ask me to solve this by using the rule of sine and cosine: Let $\alpha$, $\beta$ and $\gamma$ be the angles…
GreatDuke
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How to prove $2\sin^2(80^\circ) - \sin(70^\circ)=1$?

$2\sin^2(80^\circ) - \sin(70^\circ) = 1$ I can verify with a calculator that equality does hold, however how do I prove it with trigonometric identities?
orange orange
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what is the difference between $(\cos A)^2$ and $\cos^2 A$

$$ (|\cos A|)^2 \qquad\text{and}\qquad \cos^2 A $$ For example if $\cos A = 0.5$, and $0.5 \times 0.5 = 0.25$, are here some difference in the notations or are they equal?
Человек и PHP
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why $\tan x = \frac{\sin x}{\cos x}$? and not $\tan x$ = opposite/adjacent?

we know that $\tan x =\left(\frac{\text{opposite}}{\text{adjacent}}\right)$, but sometimes I see that $\tan x = (\frac{\sin x}{\cos x})$, is that the same thing or why it is different sometimes? cause when $\tan x…
Человек и PHP
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Simple trigonometry equation

The previous class we were doing trigonometry exercises. Before the class finished, our teacher wrote exercises on the table. I am stuck with the following one: $$ \cos(2x) + 1 + 3\sin x = 0 $$ I have come up with this: $$ 1= \sin^2 x + \cos^2…
Triak
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Evaluate $\sin\left(-\frac{\pi}{6} + \frac{1}{2}\arccos\left(\frac{1}{3}\right)\right)$.

My task is to evaluate $$\sin\left(-\frac{\pi}{6} + \frac{1}{2}\arccos\left(\frac{1}{3}\right)\right).$$ I think I've gotten most of the way there but I keep running into trouble... any suggestions?
Flack Ninja
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How do I solve the trigonometric equation $1 - \sin^2x - \cos(2x) = \frac{1}{2}$?

Solve for $x$ when $1-\sin^2x - \cos 2x = \dfrac{1}{2}$. I can' t change it into a form I can work with. It is rather complicated.
user242559
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$\cot^{-1}\frac{y}{\sqrt{1-x^2-y^2}} = 2\tan^{-1}\sqrt{\frac{3-4x^2}{4x^2}} - \tan^{-1}\sqrt{\frac{3-4x^2}{x^2}} $

Express $$\cot^{-1}\frac{y}{\sqrt{1-x^2-y^2}} = 2\tan^{-1}\sqrt{\frac{3-4x^2}{4x^2}} - \tan^{-1}\sqrt{\frac{3-4x^2}{x^2}} $$ as a rational integral equation between x and y. This is what I've done: Let $$t =…
Shaurya Gupta
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Proving simple trigonometric identity

I need help with this one $$ \frac{\sin^2 \alpha}{\sin\alpha - \cos\alpha} + \frac{\sin\alpha + \cos \alpha}{1- \mathrm{tan}^2\alpha} - \cos\alpha = \sin \alpha $$ I tried moving sin a on the other side of the eqation $$ \frac{\sin^2…
user241217
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Purely algebraic proof of the trigonometric inequalities

While calculating various limits of trigonometric functions, one must resort to the squeeze theorem which is founded on the inequalities $$1 > \frac{\sin x}{x} > \cos x$$ for some "small" $x$. These inequalities are, however, always (to my…
user54031
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Write the complex number in trigonometric form (homework question)

Write the complex number in trigonometric form, once using degrees and once using radians. Begin by sketching the graph to help find the argument θ. (Do not use cis form.) $$−1 + i$$ My work: I graphed $x = -1$ and $y = 1$ $$z=r= \sqrt{ x^2 +…
redhound2441
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Verifying - Trigonometry Homework

I have a major problem with verifying a Trigonometric identity. (My teacher couldn't really get it, so I would like to find it just in case it appears on a test) The problem goes like this: $$\sin(\theta) + \cos(\theta) = \frac{\sin(\theta)}{1 -…
nmagerko
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