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.

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Prove that $(x^2 + y^2 - z^2)^2 = 4x^2y^2$.

We have $$x\cos \theta+y\cos \phi = -z\cos \psi \tag 1$$ $$x\sin \theta+y\sin \phi = -z\sin \psi \tag 2$$ $$x\sec \theta+y\sec \phi = -z\sec \psi \tag 3$$ and we have to prove that $$(x^2 + y^2 - z^2)^2 = 4x^2y^2$$ squaring (1) & (2), and adding…
Sayantan Koley
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Area of a circle as it changes to an ellipse when viewing at different angles

Sorry for the long paragraph, I'm not too good at describing things: Firstly, to explain the situation, I need to know how the area of a circle can change as you view it from different angles. For example, if you were viewing a circle at a 0 degree…
J. Doe
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Hints to prove $\tan^2{\theta}=\tan{A}\tan{B}$, given $\frac{\sin{(\theta + A)}}{\sin{(\theta + B)}} = \sqrt{\frac{\sin{2A}}{\sin{2B}}}$

I need some hints on solving this trigonometry problem. Problem If $\dfrac{\sin{(\theta + A)}}{\sin{(\theta + B)}} = \sqrt{\dfrac{\sin{2A}}{\sin{2B}}}$, then prove that $\tan^2{\theta}=\tan{A}\tan{B}$. I tried to expand the left hand side of the…
Muhamad Abdul Rosid
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Geometric identity, cannot show equivalence using trigonometric identities

clearly $$(x+a \cos\theta)^2+(y-a \sin\theta)^2=b^2$$ expanding and using the Weierstrass substitution we find that $$\theta= 2 \arctan \frac{\left( 2ay- \sqrt{ 4a^2y^2 - ( (x-a)^2+y^2-b^2)( (x+a)^2+y^2-b^2) }\right)}{(x-a)^2+y^2-b^2} $$ if we…
Fabio Dalla Libera
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Trig equation $\sin(x)+\sin(3x)=0$, the answers are given in a factored form

$\sin(x)+\sin(3x)=0$ So to solve this I tried the following: -First I transformed this expression from sum to product because it equals zero $\sin(x)+\sin(3x)=0 => 2\sin(\frac{x+3x}{2})\cdot\cos(\frac{x-3x}{2})=0$ -Second part I got to…
L.B
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Evaluating the products $\prod\limits_{k=1}^{[n/4]} \tan \dfrac{k\pi}{n}$

I'm reseaching a problem that lead me to evaluating the product $$\prod\limits_{k=1}^{[n/4]} \tan \dfrac{k\pi}{n}=?,$$ I can evaluate $$\prod\limits_{k=1, k\neq \frac{n}{2}}^{n}\vert \tan \dfrac{k\pi}{n}\vert=\left\{ \begin{array}{l}n, n \text{ is…
J.Simpson
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Prove that the minimum value of $\sin^n(x)+\cos^n(x)$ is $\frac{1}{2^{\frac{n}{2}-1}}$, if n is even

A few days back, I discovered a relationship between $\sin^n(x)+\cos^n(x)$, when n is even. Its minimum value was always $\frac{1}{2^{\frac{n}{2}-1}}$. I tried to prove this, and to extend it to the case where $n$ was odd, but I failed to do so. Can…
Abhigyan
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Prove that $2\cdot \cos \frac{72}{2}\cdot \cos \frac{24}{2}+2\cdot \sin \frac{96}{2}\cdot \sin \frac{72}{2}=0.5$. Additional data added

To solve it I have tried some options, where in one of them I applied product to sum formulas, which seemed to be very helpful, but didn't get the answer. Used these formulae: $$\cos(a+b)+ \cos(a-b)=2\cdot \cos(a)\cdot…
user36339
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camel hump with trigonometry

sorry if this is a fool question but i use math mainly for playing and i'm not a math guru. i want to understand how get a camel hump with trigonometry, with some parameters. changing parameters i would like to set one hump more higher than the…
nkint
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Determine exact values using angle sum and difference identity (trig)

Ok so we have just learnt the basic angle sum and difference identities: $$\begin{array}{l}\cos \left( {A \pm B} \right) = \cos A\cos B \mp \sin A\sin B\\\sin \left( {A \pm B} \right) = \sin A\cos B \pm \cos A\sin B\\\tan \left( {A \pm B} \right) =…
A.Mahony
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Solve $2\cos^2{x}=\sqrt{3}\sin{2x}$.

Problem: Solve $2\cos^2{x}=\sqrt{3}\sin{2x}$ and give the sum of all the solutions in the interval $0\leq x\leq2\pi.$ Attempt: Using the fact that $\sin{2\theta}= 2\cos{\theta}\sin{\theta}$ on the RHS I get…
Parseval
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Expand $\cos^n (x)$ in terms of $\cos{kx}$, $k=1,\dots,n$.

Is it possible to expand $\cos^n(x)$ as a function of $\cos(kx)$? i.e. if $\cos^n (x)$ can be expanded as the following series $$ \cos^n (x) = \sum_{k=0}^{n} a_k \cos{kx} $$ then what are the constants $a_k$? If not, is there any way to recover…
Chris
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Find the value of $\sin (\frac{\pi}{10})$ geometrically.

The title says it. I had the idea to use the $2\theta$ and $5\theta$ formulae but they are not geometric... The question asks for an algebraic solution as well. Any ideas?
Logan Luther
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Geometric proof for trigonometric angle sum formulas.

For some reason I'm starting to get nit picky about everything I have learned. In Geometry/ Algebra I learned how to prove the angle sum formulas, which follow from this picture (and a slightly different picture with $a$ and $b$ instead of $\beta$…
Ahmed S. Attaalla
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Find the area of given triangle

In a triangle $ABC$, where $a=8$, $c=1$ and $\cos (A-C) ={16\over 65}$. Where $(a,b,c)$ are sides and $(A,B,C)$ are angles. How can we find the area of the triangle $ABC$? A hint will be appreciated.
user354545
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