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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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Solve $\sin x = 1 - x$

How would you be solve sin x = 1 - x, without drawing the graph and manually measuring the point of intersection?
astiara
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The number of solutions to $\cos^2{x}=\cos{2x}.$

How can one quickly find out the number of solutions to $$\cos^2{x}=\cos{2x}, \ \ \ 0 \leq x \leq 2\pi \ ? $$ I rewrote the equation as $$\cos2x=\cos^2{x}-\sin^2{x} \Longleftrightarrow -\sin^2{x} = 0 \Longleftrightarrow \sin{x}=0.$$ So, the equation…
Parseval
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Proving $n^2 \csc^2(nx) = \sum_{k=0}^{n-1}\csc^2\left(x+ k \frac{\pi}{n}\right)$ (without calculus?)

I recently came across the following trigonometric identity in a test: $$ n^2 \csc^2(nx) = \sum_{k=0}^{n-1}\csc^2\left(x+ k \frac{\pi}{n}\right) $$ The question was to prove the result for any natural number $n$. What would be a good way to…
Newton
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Trigonometry Inequality

This is the first time I'm posting here. If you can also tell me how to format this like a pro, I'll be very grateful. 1st question: Prove the following inequality: $$0^{\circ} < a, b, c < 180^{\circ}$$ $$\sin a \times \sin b \times \sin c \le…
yuvalz
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What is the Arctangent of Tangent?

This is probably just me misunderstanding trig properties. I remember from my trig days that $\tan(\arctan(x)) = x$ But I can't remember if that holds the other way. Can someone help me out? Is this valid: $$\arctan(\tan(\theta)) = \theta$$
Jonathan Mee
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Solving $ \sin x + \sqrt 3 \cos x = 1 $ - is my solution correct?

I have an equation that I'm trying to solve: $$ \sin x + \sqrt 3 \cos x = 1 $$ After pondering for a while and trying different things out, this chain of steps is what I ended up with: $$ \sin x + \sqrt 3 \cos x = 1 $$ $$ \sin x = 1 - \sqrt 3 \cos x…
tereskopu
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What is $\sin(nx)$ iteration in terms of $\sin A$ and $\cos A$?

I want to use sum angle formulas, $\sin(A+B)=\sin A\cos B+\cos A\sin B$ to get for any angles, $\sin(nA)$ in terms of powers of $\sin A$ and $\cos A$. I know there are other ways, but I want to use trigonometry and iteration on that. The goal is to…
koe
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Help with this trigonometry problem

Prove the given identity. $$\left(\sin\frac {9\pi}{70}+ \sin\frac {29\pi}{70} - \sin\frac {31\pi}{70}\right) \left(\sin\frac {\pi}{70}-\sin\frac {11\pi}{70} - \sin\frac {19\pi}{70}\right) =\frac {\sqrt {5} -4}{4}$$ Please help, I could not gather…
pi-π
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Why do hyperbolic "trig" functions seem to be encountered rarely?

Hyperbolic "trig" functions such as $\sinh$, $\cosh$, have close analogies with regular trig functions such as $\sin$ and $\cos$. Yet the hyperbolic versions seem to be encountered relatively rarely. (My frame of reference is that of someone with…
Tom Au
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If $\sin x + \sin y = 1$ and $\cos x + \cos y = 0$, solve for $x$ and $y$

$\sin x + \sin y = 1$ $\cos x + \cos y = 0$ Any valid pair of $(x, y)$ is fine, as the restrictions on the board in the image below are obscured. I got the question from chapter 26 of a comic called Yamada-kun. How can I solve this equation?
Jack Pan
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Why isn't math on the sine of angles the same as math on the angles in degrees?

I noticed something just now. This is probably a stupid question, but I'm going to ask it anyway. Because when I discover that my understanding of a topic is fundamentally flawed, I get nervous. Basically I'm suppose to show that the angle marked…
Algific
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What are the practical applications of this trigonometric identity?

On various occasions people have asked here how to prove (but that is NOT the question here) that $$ \text{if } \alpha+\beta+\gamma = \pi \text{ then } \frac{\sin(2\alpha) + \sin(2\beta) + \sin(2\gamma)} 2 = 2\sin\alpha\sin\beta\sin\gamma. $$ I…
Michael Hardy
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Prove $\cos 3x =4\cos^3x-3\cos x$

How would I solve the following double angle identity. $$\cos 3x =4\cos^3x-3\cos x $$ I know $\,\cos 3x = \cos(2x+x)$ So know I have $\,\cos 2x +\cos x \,$ , Which is $\,(2\cos^2x-1)\cos x$ But I am not sure what to do next.
Fernando Martinez
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Why is the range of inverse trigonometric functions defined in this way?

My question is really simple, why is the range of the $\sin^{-1}(x)$, $\cos^{-1}(x)$ and $\tan^{-1}(x)$ defined as $[-\pi/2,\pi/2]$, $[0,\pi]$ and $[-\pi/2,\pi/2]$ respectively? Is there some particular reason? We could choose another range for each…
user42912
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Solve the equation $27 \sin(x) \cdot \cos^2(x) \cdot \tan^3(x) \cdot \cot^4(x) \cdot \sec^5(x) \cdot \csc^6(x) = 256$.

Solve the equation $27 \sin(x) \cdot \cos^2(x) \cdot \tan^3(x) \cdot \cot^4(x) \cdot \sec^5(x) \cdot \csc^6(x) = 256$. I was hoping some things would cancel out when I expanded this but nothing. I think using inequalities will help.
user19405892
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