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.

29665 questions
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Find $\cos{A}+\cos{B}$

In $\Delta ABC$,if $$\cos{C}\cdot(\sin{A}+\sin{B})=\sin{C}\cdot\cos{(A-B)}$$ Find $\cos{A}+\cos{B}$ Thus…
user225250
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What is this expression means? $\sin^{-2}x$

I know this is very silly question but I didn't know how to research it in google. Please bare with me on this one. I have two thoughts: $$\sin^{-2}x = \frac{1}{\sin^2x}$$ I think this cannot be possible. Or is it: $$\sin^{-2}x = \arcsin^2x$$ I…
Bora
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Prove that the trigonometric equation has no solution

Prove that the trigonometric equation $$\frac{\sin^3 x}{1-\sin x}+\frac{\cos^3 x}{1-\cos x}=-1$$ has no solution. I tried applying $T2's$ lemma to contradict but could only do so for the first and third quadrant values if $x$. There must be some…
user167045
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What are the ways to solve trig equations of the form $\sin(f(x)) = \cos(g(x))$?

if I have the following trig equation: $$\sin(10x) = \cos(2x)$$ I take the following steps to solve it: I rewrite $\cos(2x)$ as $\sin\left(\frac{\pi}{2} + 2x\right)$ or as $\sin\left(\frac{\pi}{2} - 2x\right)$ cause $\sin\left(\frac{\pi}{2} -…
user3019105
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$ \cos ^2\left(x\right)+\cos ^2\left(2x\right)+\cos ^2\left(3x\right)=\frac{3}{2} $

$$ \cos ^2\left(x\right)+\cos ^2\left(2x\right)+\cos ^2\left(3x\right)=\frac{3}{2} $$ How can I solve this one, I mean I get something like this: $-3+\left(-1+2\cos ^2\left(x\right)\right)^22+2\left(-3\cos \left(x\right)+4\cos…
random_name_test
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Prove $\sin{2\theta} = \frac{2 \tan\theta}{1+\tan^{2}\theta}$

How to prove: $$\sin{2\theta} = \frac{2 \tan\theta}{1+\tan^{2}\theta}$$ Help please. Don't know where to start.
Pheter
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Finding the period of $f(x) = \sin 2x + \cos 3x$

I want to find the period of the function $f(x) = \sin 2x + \cos 3x$. I tried to rewrite it using the double angle formula and addition formula for cosine. However, I did not obtain an easy function. Another idea I had was to calculate the zeros and…
user30523
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$\sin(x^2)$ in terms of $\sin(x)$ and $\cos(x)$

One of my students asked me "Can you write $\sin(x^2)$ in terms of $\sin(x)$"? I said I'd think about it. Having thought about it for a while, I now know that I definitely don't know the answer! Lets relax the question to "...$\sin(x^2)$ in terms of…
AlanGIC
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Show that $ \frac{\cos5x + \cos4x} {1-2\cos3x} = -\cos2x -\cos x $

The Question reads - $$ \frac{\cos5x + \cos4x} {1-2\cos3x} = -\cos2x -\cos x $$ I tried using the obvious approach by converting $5x , 4x $ and $ 3x$ to either $2x$ or $x$ but all that seemed to do was to further complicate the fraction. Any hints…
MayankJain
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Finding the least positive root

How to find the least positive root of the equation $\cos 3x + \sin 5x = 0$? My approach so far is to represent $\sin 5x$ as $\cos \biggl(\frac{\pi}{2} - 5x\biggr)$ then the whole equation reduces to $$2\cos \biggl(\frac{\pi}{4} - x\biggr)\cdot \cos…
Quixotic
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Continued product in $\sin$ series

Find the value of the product $$(\sin 1°)(\sin 3°)(\sin 5°)\ldots(\sin 89°)$$ I tried multiplying and dividing by $2$ and then combining and then converting into cosine, but doesn't work out.
user167045
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How would you solve for all solutions of $\sin(2x)=\cos(3x)$ algebraically?

So my buddy and I (both HS Math teachers) have been messing around with a question about finding all solutions of a "co-function" equation like the one above. The typical HS questions asks students to find the solution in Q1 by solving $2x + 3x =…
Mike Andrejkovics
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If $A,B>0$ and $A+B = \frac{\pi}{3},$ Then find Maximum value of $\tan A\cdot \tan B$.

If $A,B>0$ and $\displaystyle A+B = \frac{\pi}{3},$ Then find Maximum value of $\tan A\cdot \tan B$. $\bf{My\; Try::}$ Given $$\displaystyle A+ B = \frac{\pi}{3}$$ and $A,B>0$. So we can say $$\displaystyle 0< A,B<\frac{\pi}{3}$$. Now taking $\tan $…
juantheron
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How can you rigorously define trigonometric functions without series or integrals?

In middle school, cos and sin were defined with angles, and in high school, with the length of an arc of the unit circle. But angles, are defined with cos and you need integrals to define the length of an arc! And the definition with series is a bit…
Soc
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How do you square $\sin θ\,$?

Is $(\sinθ)^2=\sin^2θ$ or $(\sinθ)^2=\sin(θ^2)$ or $(\sinθ)^2=\sin^2(θ^2)$ Can you explain your answer, regards Tom. Also, does your answer work for $\cos$ and $\tan$?
Thomas Winkworth
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