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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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A trigonometry equation problem

Let $a,b \in \left(0,\frac{\pi}{2}\right)$, satisfying $$ \frac{1-\cos{2a}}{1+\sin{a}}+\frac{1+\cos{2a}}{1+\cos{b}}=\frac{1-\cos{(2a+2b)}}{1+\sin{(a+2b)}} $$ Prove that: $$ a+b=\frac{\pi}{2} $$
Golbez
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Prove $\frac{1-\tan(x/2)}{1+\tan(x/2)}=\frac{1-\sin x}{\cos x}$

Can anyone offer please help me solve the following trig identity. $$\frac{1-\tan(x/2)}{1+\tan(x/2)}=\frac{1-\sin x}{\cos x}=\frac{\cos x}{1+\sin x}$$ My work thus far has been on the left most side. I did…
James El Cholo
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Solving $2\cos\left(2\theta\right) = \sqrt{3}$

I have a question on this test review problem (that will help us on a test), and I have no clue what it's asking. We're learning trigonometry, (Analytic Trigonometry), like about the unit circle, inverse trig functions, etc... And I encounter this…
amanuel2
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Finding the values of $\cos \frac{n\pi}{2}$ and $\sin \frac{n\pi}{2}$.

i know that the values of $\cos n\pi=(-1)^{n}$ and $\sin n\pi=0$. Now i want to know that what is the general expressions of $\cos \frac{n\pi}{2}$ and $\sin \frac{n\pi}{2}$.
Vishal
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Trigonometric equation $\cot(x)\csc(x)=2\sec^2(2x)$

I come up with the equation: $\cot(x)\csc(x)=2\sec^2(2x)$ This looks extremely simple but I am not able to come up with a simple solution for $x$. I work out that $4\cos^5(x)-4\cos^3(x)+2\cos^2(x)+\cos(x)-2=0$ Am I on the right track or is there…
Mc Cheng
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$3\sin^2x=\cos^2x;$ $ 0\leq x\leq 2\pi$ Solve for $x$

$3\sin^2x=\cos^2x;$ $0\leq x\leq 2\pi$ Solve for $x$: I honestly have no idea how to start this. Considering I'm going to get a number, I am clueless. I have learned about $\sin$ and $\cos$ but I do not know how to approach this problem. If anyone…
Austin Broussard
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What is the integral value of $\frac{\tan 20^\circ+\tan40^\circ+\tan80^\circ-\tan60^\circ}{\sin40^\circ}$?

I have tried possibly all approaches. I first expressed $80$ as $60+20$ and $40$ as $60-20$ and then used trig identities.I later used conditional identities expressing $\tan 20^\circ+\tan40^\circ+\tan120^\circ$ as $\tan 20^\circ \tan40^\circ…
aakanksha reddy
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Solve $\sin x = x - 2 \pi/3$

What is $x$ if $\sin x = x - 2 \pi/3$? The answer is $x \approx 2.61$ but how do I work that out (without Taylor series - this is homework for 10th grade)? Thanks.
user46234
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How do you prove that $\arcsin( \frac{1}{2} \sqrt{2-\sqrt {2-2x}})=\frac{\pi}{8} + \frac{1}{4} \arcsin x$?

I have the task to prove that $$ \arcsin( \frac{1}{2} \sqrt{2-\sqrt {2-2x}})=\frac{\pi}{8} + \frac{1}{4} \arcsin x ,\left|x\right|\le 1 $$ I do not have any ideas from where I should start. Can anyone help me solve it?
Diyan Bogdanov
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Verify the following trigonometric identity.

Verify the following trig identity: $$\sin(3\theta)-\sin\theta = 2\cos(2\theta)\sin\theta$$ Here is my work so far. $\sin(3\theta)-\sin\theta = 2\cos(2\theta)\sin\theta$ LHS:$$\sin(\theta+2\theta)-\sin\theta$$ $$\sin\theta…
McB
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Find the side length of triangle ABC;

Let ABC be an equilateral triangle, and let P be a point in the interior of the triangle. Given that PA = 3, PB = 4, and PC = 5, find the side length of ABC. Relatively simple problem I think, but I can't quite get the right way to solve this.…
mathflair
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Express $\arcsin(x)$ in terms of $\arccos(x)$. Solve the equation 2 arctan x=arcsin x + arccos x

Express $\arcsin(x)$ in terms of $\arccos(x)$. Using the same, solve the equation $$ 2\,\tan^{-1}x = \sin^{-1} x + \cos^{-1} x $$ I'm not sure if I am on the right track, but here is what i did: $$\sin\left(\frac{\pi}{2}-x\right) =…
Math Solver
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A trigonometric identity: $(\sin x)^{-2}+(\cos x)^{-2}=(\tan x+\cot x)^2$

I've been trying to prove it for a while, but can't seem to get anywhere. $$\frac{1}{\sin^2\theta} + \frac{1}{\cos^2\theta} = (\tan \theta + \cot \theta)^2$$ Could someone please provide a valid proof? I am not allowed to work on both sides of the…
user26649
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Validity Michael Hardy's proof of Pythagoras Theorem using differentials

A proof of the pythagorean theorem has been published by Mike Hardy during 1988 in Mathematical Intelligencer (Hardy, Michael, "Pythagoras Made Difficult". Mathematical Intelligencer, 10 (3), p. 31, 1988.). The proof can be found at…
Κώστας Παπαστεργίου
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Solve trigonometric equation $ \cot x + \cos x = 1 + \cot x \cos x $

Solve trigonometric equation: $$ \cot (x) + \cos (x) = 1 + \cot (x) \cos (x) $$ I tried to multiply both sides with $\sin x$ (which I'm not sure if I can multiply with sin).
Gjekaks
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