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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What is $ \sin(x)+\sin(x−π)+\sin(x+π) $?

So I have this trig question: $ \sin(x)+\sin(x−π)+\sin(x+π) = $ _____ The answer is $- \sin(x)$ I can't figure out how to solve it. Any help?
Arkilo
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show this identity with trigometric

I sent a post earlier. Follow is an original problem. I got an error identity from a previous calculation error. Now there should be no problem. Problem:: let $x,y\in (0,\dfrac{\pi}{2})$. show…
math110
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If $\sec\theta=-\frac{13}{12}$, then find $\cos{\frac{\theta}{2}}$, where $\frac\pi2<\theta<\pi$. The official answer differs from mine.

Given $\sec\theta=-\frac{13}{12}$ find $\cos{\frac{\theta}{2}}$, where $\frac\pi2<\theta<\pi$. If the $\sec\theta$ is $-\frac{13}{12}$ then, the $\cos \theta$ is $-\frac{12}{13}$, and the half angle formula tells us that $\cos{\frac{\theta}{2}}$…
dstarh
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Tips for understanding the unit circle

I am having trouble grasping some of the concepts regarding the unit circle. I think I have the basics down but I do not have an intuitive sense of what is going on. Is memorizing the radian measurements and their corresponding points the only way…
Matt
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If $\tan{\frac{x}{2}}=\csc x - \sin x$, then find the value of $\tan^2{\frac{x}{2}}$.

If $\tan{\frac{x}{2}}=\csc x - \sin x$, then find the value of $\tan^2{\frac{x}{2}}$. HINT: The answer is $-2\pm \sqrt5$. What I have tried so far: $$\tan{\frac{x}{2}} = \frac{1}{\sin x}-\sin x$$ $$\tan{\frac{x}{2}} = \frac{1-\sin^2 x}{\sin…
user642405
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Solving $2\sin\theta\cos\theta + \sin\theta = 0$

The question is to solve the following question in the range $-\pi \le \theta \le \pi$ $$2\sin\theta\cos\theta + \sin\theta = 0$$ I missed the obvious sin factorisation so proceeded as below. I see the correct solutions should be $\pm2/3\pi$ and…
Joseph
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How do I prove that $\cos\left(2x\right)=1-2\sin^2\left(x\right)$?

While trying to solve the equation $\sin\left(x\right)=\cos\left(2x\right)$, a user on this forum suggested that I turn the equation into a quadratic form by converting $\cos(2x)$ using the identity…
Daniel Oscar
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Trigonometry problem $\sin100^\circ+\cos70^\circ\over\cos80^\circ-\cos20^\circ$

What is the value of: $$\sin100^\circ+\cos70^\circ\over\cos80^\circ-\cos20^\circ$$ I've done trigonometry in my earlier years of high school but I forgot a lot of rules. This is where I'm stuck on this…
Aleksa
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Is $\cot x = \tan (π/2 - x) $ true for any angle $x$?

Is $$\cot x = \tan \Big(\frac{π}{2} - x\Big)$$ true even when $x$ is not an acute angle ?
anonymous
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Solve for $\theta$: $x = \theta - \sin\theta$

Solve for $\theta$: $$x = \theta - \sin\theta$$ Is this type of isolation a matter of identities? If so, which one(s)?
user101434
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Showing that $\ln(\sec 3t+\tan 3t)=2\tanh^{-1}(\tan(3t/2))$

I've been trying to show that $$\ln(\sec 3t+\tan 3t)=2\tanh^{-1}(\tan(3t/2))$$ I used the identity $$\tanh^{-1}x=\frac12\ln\frac{1+x}{1-x}$$ to…
David
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Trigonometrical Problem

I think this is a bit odd but I am juggling since hours with $\sin$, $\cos$, $\tan$ and other stuff to proof a formula, but I can't do it. Slowly I am thinking that this formula is wrong. Maybe there is some expert who could tell me if I am right. I…
Miau
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Closed form for $\sin(n\arctan(x))$

Is there a closed form for the function $\sin(n\arctan x)$, perhaps where $n$ is restricted to being an integer, or if not, perhaps some special integers (such as triangular numbers or some other figurate numbers)? From playing around with a few…
Hobbyist
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How to show that $\cos(\sin^{-1}(x))$ is $\sqrt{1-x^2}$?

How to show that $\cos(\sin^{-1}(x))$ is $\sqrt{1-x^2}$? I remember having to draw a triangle, but I'm not sure anymore.
Bas
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Find the values of $x$ and $y$ that satisfies $\sin(x+y)=\sin x+\sin y$.

I know that in general the following equality does not hold: $\sin(x+y)=\sin x + \sin y$. However, I have been looking for specific values of $x$ and $y$ that satisfies the given equation. This is what I have done so far: $\sin(x+y)=\sin x\cos y +…
user573497
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