Mathematics Stack Exchange
Mathematics Stack Exchange

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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Finding $\alpha$ such that $\;\tan\alpha=\frac{8\sqrt3\cos^240^\circ+8\sqrt3\cos20^\circ-2\sqrt3}{(8\cos^240^\circ-1)(8\cos20^\circ-1)-3}$

I have to find $\alpha$ (in degrees) so it is satisfied $$\tan{\alpha}=\frac{8\sqrt{3}\cos^{2}40^{\circ}+8\sqrt{3}\cos20^{\circ}-2\sqrt{3}}{\left(8\cos^{2}40^{\circ}-1\right)\left(8\cos20^{\circ}-1\right)-3}$$ I tried to write…
JohnB
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Find the maximum perimeter of a right angled triangle with hypotenuse 1

This is a question from the grade 7 Math Competition. I can solve it by considering one of the angles, say $\theta$, besides the right angle. We have the perimeter given by $1+\cos\theta+\sin\theta = 1+\sqrt{2}\cos(\theta-\pi/4 )$ and the rest is…
Anson NG
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prove that $\cos x,\cos y,\cos z$ don't make strictly decreasing arithmetic progression

let $x,y,z\in R$,and such that $$\sin y-\sin x=\sin z-\sin y\ge 0 $$ show that: $$\cos x,\cos y,\cos z$$ don't make strictly decreasing arithmetic progression my idea: we have $$2\sin y=\sin x +\sin z\cdots\cdots\tag 1$$ and assume that,there…
math110
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$\arctan$ identities: converting $c \cdot \arctan(\frac{n}{d}) \iff \arctan(\frac{n'}{d'})$

If one has an expression of the form $c \cdot \arctan(\frac{n}{d})$, it can be converted to an equivalent expression of the form $\arctan(\frac{n'}{d'})$ fairly easily, using $c \cdot \arctan(\frac{n}{d}) = \underbrace{\arctan(\frac{n}{d}) +…
primo
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Solve $\sin(x)+2\sin(x)\cos(x)=\pi/4$

Is it possible to solve (not approximate) the following trigonometric equation by hand? $$\sin(x)+2\sin(x)\cos(x)=\pi/4.$$
Ovi
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Determining whether an angle is between two given angles on the unit circle

I am trying to find a way to determine whether an angle is between two given angles where all angles are provided as vectors on the unit circle i.e.: $\mathbf{a}=(\cos(\theta),\sin(\theta))$ Note that by inbetween I mean on the arc of the smaller…
Plog
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Show that $\frac{3\> + \>\cos x}{\sin x}$ cannot have any value between $-2\sqrt2$ and $2\sqrt2$

Show that $$\dfrac{3+\cos x}{\sin x}\quad \forall \quad x\in R $$ cannot have any value between $-2\sqrt{2}$ and $2\sqrt{2}$. My attempt is as follows: There can be four cases, either $x$ lies in the first quadrant, second, third or fourth:- First…
user3290550
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Show that for all $(x,y)$ there exists $(r,\theta)$...

Given a problem wherein $(x,y) \in \mathbb{R}^2$, I often transform to polar coordinates by introducing the assumption that $(r,\theta)$ satisfy $x = r\cos \theta, y = r\sin \theta.$ Of course, it's only safe to introduce assumptions if you have an…
goblin GONE
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If $u=\sqrt {a\cos^{2}x+b\sin^{2}x} + \sqrt {b\cos^{2}x+a\sin^{2}x}$ , find the maximum and minimum value of $u^2$.

If $u=\sqrt {a\cos^{2}x+b\sin^{2}x} + \sqrt {b\cos^{2}x+a\sin^{2}x}$ , find the maximum and minimum value of $u^2$. This problem was bothering me for a while. The minimum value of $u $ seemed relatively easy to find by using AM-GM followed by…
shsh23
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Maximizing $ 3^{\sin^2{\theta}} \cdot 27^{\cos^2{\theta}} + 8^{\sin{\theta}}\cdot 16^{\cos{\theta}} $

Find the maximum value of $$ 3^{\sin^2{\theta}} \cdot 27^{\cos^2{\theta}} + 8^{\sin{\theta}}\cdot 16^{\cos{\theta}} $$ Where does maximum of the expression occur? I can find maxima of individual terms easily, but since they occur at different…
George carlin
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In the equation $x\cos(\theta) + y\sin(\theta) = z$ how do I solve in terms of $\theta$?

In the equation $$x\cos(\theta) + y\sin(\theta) = z,$$ how do I solve in terms of $\theta$? i.e $\theta = \dots$.
user9624
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$\sqrt2 x^2 - \sqrt3 x + k = 0$ with solutions $\sin \theta , \cos \theta$

$\sqrt2 x^2 - \sqrt3 x + k = 0$ with solutions $\sin \theta , \cos \theta $, $\enspace0\leq\theta\leq2\pi$. $(x-\sin \theta)(x-\cos \theta)=0$ $(\sin \theta + \cos \theta) = \sqrt3/ \sqrt2$ $(\sin \theta \cdot \cos \theta) = k/\sqrt2$ But how to…
Dini
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Can this equation be solved? $x+\sin(x)=\frac{11\pi}{48}$

So I was revisiting an older problem and seeing if I could solve it in a different way. I boiled the equation down to this: $$x+\sin(x)=\frac{11\pi}{48}$$ I can't imagine how to isolate x, and a number of computer solvers also broke down in the…
Abel Palmer
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Can one simplify $\arctan(a\tan(x))$?

We know that $\arctan(\tan(x))=x$ when $x$ lies between $-\pi/2$ and $+\pi/2$; but do you know a way to transform the expression $\arctan(a\tan(x))$, where $a$ is a real number between $0$ and $1$? I thought $a$ could be transformed with…
Andrew
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Calculate the height of a distant object using estimated angles from two different points.

I knew how to do this long ago, found the exact problem in my old trig book, but I can't seem to work it out. Say I'm at an unknown distance from a mountain, called point P, and I estimate the angle of elevation to the top of the mountain is 13.5…
Nocturno
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