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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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Functional inverse of $\sin\theta\sqrt{\tan\theta}$

What is the functional inverse of $f(\theta) = \sin\theta\sqrt{\tan\theta}$? Or, equivalently, what is the inverse of $$f(\theta)=\sin^2\,\theta\tan\,\theta=\frac{\sin^3\,\theta}{\cos\,\theta}$$ It comes from a physics setup involving two…
algorithmshark
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Finding the maximum of $5\sin x+4\sin 2x$

How does one find the maximum value of $$ 5\sin(x)+4\sin(2x) $$ without using calculus?
TSP1993
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In a triangle ABC, (b + c) cos A + (c + a) cos B + (a + b) cos C is equal to

In a triangle $ABC$ $$(b + c)\cos A + (c + a)\cos B + (a + b)\cos C=?$$
burm1
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if $a$ is are real number that $a \neq 0$, and $\cos x = \sqrt{\frac{\cot x}{\cot x -a^2}}$, $x$ is on which trigonometric quadrants?

if $a$ is are real number that $a \neq 0$, and $\cos x = \sqrt{\frac{\cot x}{\cot x -a^2}}$, $x$ is on which trigonometric quadrants? Things I have done so far: this problem is mostly different from that I previously solved.My Idea was to…
user2838619
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Trigonometry Identity: $\tan \theta\sin \theta + \cos \theta = \sec \theta$

Sorry if my question seems too simple. I cannot find a proof and my text book does not provide one either. I am supposed to prove: $$\tan \theta \times \sin \theta + \cos \theta = \sec \theta$$ I know that $\sec = \frac{1}{\cos\theta}$. But I do not…
bman
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Why is $\sin \theta$ just $\theta$ for a small $\theta$?

When $\theta$ is very small, why is sin $\theta$ taken to be JUST $\theta$?
pblead26
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Solving an equation with arctan and arcsin

I am trying to do what I think a problem with a simple answer. Here are the two equations I have resolved the problem down to: $$\angle A = \arctan \frac{28}{x}$$ and $$\angle A = \arcsin \frac{1}{12-x}$$ Is it not right then that I can make…
jeff
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Expressions of $\sin \frac{A}{2}$ and $\cos \frac{A}{2}$ in terms of $\sin A$

I am trying to understand the interpretation of $\sin \frac{A}{2}$ and $\cos \frac{A}{2}$ in terms of $\sin A$ from my book, here is how it is given : We have $ \bigl( \sin \frac{A}{2} + \cos \frac{A}{2} \bigr)^{2} = 1 + \sin A $ and $ \bigl( \sin…
Quixotic
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Proof of trigonometric identity $\cot \theta \sec\theta= 1/ \sin\theta$

Is this trigonometric identity provable? $$\color{red}{}\;\color{navy}{\cot \theta \sec \theta = \dfrac 1 {\sin \theta}}$$ I can't seem to get passed: $\dfrac{1}{\tan\theta \cos\theta}$
xsqs
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Exact value of cosine function when $x$ is not a multiple of 3 degrees

Can the exact value of cosine function be expressed as some finite combination of integers. nth power and fundamental operations($+,-,/,\times)$. When $x$ belongs to integers (in degrees) and is not a multiple of $3$. For example…
Mathslover
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Solve trigonometric equation $\sin14x - \sin12x + 8\sin x - \cos13x= 4$

I am trying to solve the trigonometric equation $$ \sin14x - \sin12x + 8\sin x - \cos13x= 4 $$ The exact task is to find the number of real solutions for this equation on the range $[0, 2\pi]$. Thanks.
Transcendental
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Prove this identity...

$$\frac{\sin 2x}{1+\cos 2x} \times \frac{\cos x}{1+\cos x}=\tan\frac{x}{2}$$ This is what I've done: $$\frac{2\sin x \cos x}{1+\cos^2 x-\sin^2 x} \times \frac{\cos x}{1+\cos x}=$$ $$\frac{2\sin x \cos x}{2\cos^2 x} \times \frac{\cos x}{1+\cos…
A6SE
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Is there a simpler/better proof of this simple trigonometric property?

The sine function has the following nice property : for any $x,y$, we have $\sin(x)+\sin(y)=\sin(x+y)$ iff at least one of $x,y,x+y$ is $0$ modulo $2\pi$. I sketch below my current proof of it, which I find somewhat unsatisfying. Does anyone know a…
Ewan Delanoy
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Measuring diaognals without Sine Law

Lets start off with a simple right angled triangle 'abc'. (ie: use cartesian coordinates, we mark 'a' and 'b' on x,y axis, 'c' is calculated from Pythagoras therom). Now pick an arbitrary point 'o' to the right of 'c'. We then measure lengths e and…
Alvin K.
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Calculate dimensions of square inside a rotated square

A given square is rotated on its center point by 'z' degrees. A new square is added inside this at no angle, whose size is based on the perimeter of the containing square. Is there a way to calculate my black square's dimensions, given the angle…
user339946
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