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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prove that if $\sin^{2}\alpha+\sin^{2}\beta+\sin^{2}\gamma=2$ then the triangle has a right angle

prove that if $\sin^{2}\alpha+\sin^{2}\beta+\sin^{2}\gamma=2$ then the triangle has a right angle. $\alpha,\beta,\gamma$ are the angles of the triangle. I tried to use all kinds of trigonometric identities but it didn't work for me. it's to complex…
gorgi
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How to prove the identity $\frac{1}{\sin(z)} = \cot(z) + \tan(\frac{z}{2})$?

$$\frac{1}{\sin(z)} = \cot (z) + \tan (\tfrac{z}{2})$$ I did this: First attempt: $$\displaystyle{\frac{1}{\sin (z)} = \frac{\cos (z)}{\sin (z)} + \frac{\sin (\frac{z}{2})}{ \cos (\frac{z}{2})} = \frac{\cos (z) }{\sin (z)} +…
VVV
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Are there any constants other than $\pi$ that give rational or known irrational values for $\cos(\theta)$?

For example: $\cos(\frac{\pi}{3}) = \frac{1}{2}$ $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ Is there any other constant $\theta$ such that $\cos(k\theta)$ is rational or a known irrational where $k$ is not $0$ or something trivial like…
transcendental
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Exact value for $\cos 36°$

Good morning! I'm having trouble with this problem... It's just taking me forever and I'm worn out and I'm lost on how to use a double angle identity for $72=2⋅36$ The problem reads as follows An exact value for $\cos36°$ can be found using the…
user5987
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Is $\cos(x^2)$ the same as $\cos^2(x)$?

I want to know something about trigonometrical functions, is $\cos(x^2)$ the same as $\cos^2(x)$ ?
cosineofx
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prove the following trig identity

For any $x \in [0,1]$ show that $$\arcsin(x)+\arccos(x)=\frac{\pi}{2}$$ Please note that this is not a homework problem, this is something I came across that appears to be true.
Wintermute
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Closed form of $\arccos\left(\frac{2 \pi}{2^N}\right)$

Is there a closed form for $\arccos\left(\dfrac{2 \pi}{2^N}\right)$ in terms of $N \in \mathbb{Z}, N \ge 3$? I'm not super optimistic, but I'm not sure how to really start exploring the problem, either.
Jay Lemmon
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Some trigonometric formula

How to prove that $1+2(\cos a)(\cos b)(\cos c)-\cos^2 a-\cos^2 b-\cos^2 c=4 (\sin p)(\sin q) (\sin r)(\sin s)$, where $p=\frac{1}{2}(-a+b+c)$, $q=\frac{1}{2}(a-b+c)$, $r=\frac{1}{2}(a+b-c)$, $s=\frac{1}{2}(a+b+c)$. Thanks.
Richard
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Trigonometry equation $\sin(x)+\cos(x)-\tan(x)=0.4$

There's some way to find $x$ here ? $$\sin(x)+\cos(x)-\tan(x)=0.4$$
Joel
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Intersection of Trig Functions

The questions asks to find the intersections of $$f(x) = 2 \sin(x-7) + 6$$ and $$g(x) = \cos(2x-10) + 8$$ within the interval $[6,14]$. So my general strategy was, 1) equate the functions, 2) get all the $X$s on one side and 3) convert to the same…
Marty B.
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How prove $A=B=C$?

in $\Delta ABC$, such $$\sin{A}+\cos{B}+\tan{C}=\dfrac{3\sqrt{3}+1}{2}$$ prove that $$A=B=C=\dfrac{\pi}{3}$$ My try: use $$\sin{x}+\sin{y}=2\sin{\dfrac{x+y}{2}}\cos{\dfrac{x-y}{2}}$$ then …
user94270
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Trigonometry - How do I simplify this expression?

We have the expression $$ 13 \sin [ \tan ^{-1} (\dfrac{12}{5}) ] $$ Apparently the answer is 12, and I have to simplify it, and I'm assuming it means I have to show it's 12, without using a calculator. Normally I show my own work in the questions,…
Phaptitude
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Solving trigonometric equation with unknown and restricted domain

Given that $ \tan^2(\fracθ3) = 1$ and $θ\in [0, 4\pi]$ find θ. I'm not sure how to progress with the restricted domain. Here's what I've got so far: Solving for the domain $[0, 4\pi]$. $$ \tan^2(\fracθ3) = 1$$ $$ \tan(\fracθ3) = 1$$ Since $…
moss
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Why is $\arctan\frac{x+y}{1-xy} = \arctan x +\arctan y$?

Why is $\arctan\frac{x+y}{1-xy} = \arctan x +\arctan y$? It is said that this is derived from trigonometry, but I couldn't find why this is the case.
KBC
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