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.

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If $ \dfrac{a^2-b^2}{a^2+b^2} = \dfrac{\sin(A-B)}{\sin(A+B)} $, then what type of triangle is $\triangle ABC $?

In $\triangle ABC$ $ \dfrac{a^2-b^2}{a^2+b^2} = \dfrac{\sin(A-B)}{\sin(A+B)} $ then what type of triangle is $\triangle ABC $ ? My try : By componendo and dividendo $\dfrac{a^2}{b^2} = \tan A \cot B$ Not able to conclude, any help ?
AgentS
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Trigonometric identity expressing $\sec \theta+\text{cosec } \theta$ in terms of sine and cosine

$\large{\text{cosec }\theta+\sec{\theta}=\dfrac{\sin\theta+\cos\theta}{\sin\theta\,\cos\theta}}$ I know that cosecant is the inverse of sine, and secant is the inverse of cosine. However, that does not equal the right hand side of the equation. I…
xsqs
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Period of trigonometric function

What is the period of $$\frac{7\sin x + 5\cos x}{7\sin{2x} + 11\cos x}$$ What should I do here? I don't even know where to start from. Please help me by giving me a hint!! Thanks.
Gummy bears
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Trigonometry Question: find Value of.....

Find value of $3 + \cos2x + \cos4x + \cos6x - 4\cos x\cos2x\cos3x$. I tried with $\cos A + \cos B$ identity but it was not simplifying.... Help..
Sudhanshu
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Period of $\sin(x) + \cos(x)$

The period of $\sin(x)$ is $2\pi$ and $\cos(x)$ is $2\pi$. And the period of $\sin(x)+\cos(x)$ is also $2\pi$. Why it is so?
user165385
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How to find the maximum value of $12\sin x -9\sin^2x$

How to find the maximum value of $12\sin x -9\sin^2x$ ; My approach : This can be written as $-[(3\sin x -2)^2-4]$. It means that the function will be maximum when $(3\sin x-2)^2 <4$ due to negative sign outside bracket. But I am not getting how…
user108258
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Solving this trigonometric equation: $\sqrt{3} \cos x - 3 \sin x = 4 \sin 2x \cos 3x$

$$\sqrt{3} \cos x - 3 \sin x = 4 \sin 2x \;\cos 3x$$ I tried many things: opening $\sin 2x$, $\cos 3x$, simplifying LHS: $\cos(60^\circ+x)$. Nothing seems to work. Any hint?
Harshal Gajjar
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Product-to-sum trigonometry identity

I'm really not sure about this Product-to-sum identity on wiki. See: I cannot find this anywhere on the web - does anybody know a reference? Certainly the one wiki gives does not cover it. I'm guessing it is used for something like…
onepound
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How to solve this problem 4

Question $1$: Is $\frac{1}{\pi}\arccos\left(\frac{{\sqrt{2*\sqrt{2*\sqrt{2}*...n}}}}{2}\right)$ always a rational number when each$*$ is either $+$ or $-$ and $n$ may or may not be infinite? Question $2$ If its a rational number, then how is it…
Arkin
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How do you find this product?

Is there a way to find the exact value of the product $$P=\displaystyle\prod_{n=1}^{1007} \sin {\left(\dfrac{n\pi}{2015}\right)}$$
user1001001
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How to solve $0=\cos{2x}+\cos x$

I'm desperately trying to solve $$0=\cos{2x}+\cos x$$ Am I on the right track when I'm this far? $$\cos x(2\cos x+1)=1$$ I don't know where to go from here. What method can I use to further solve the equation? And how do these methods work?
user1211030
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Difference between $\arccos$ and $\cos^{-1}$

I'm trying to use inverse cosine, in the cosine rule to find an angle of a triangle when you know 3 sides. I know this formula and have it written down. However I've left my calculator and I'm having to use…
Jonathan.
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Express the cosine of triple angle $3x$ in terms of cosines of $2x$ and $x$

Show that $$\cos{3x}=2\cos{2x}\cos{x}-\cos{x}$$ I've tried adding and subtracting $\cos{x}$ from $\cos{3x}$, like this: $$\cos{3x}+\cos{x}-\cos{x}$$ so I get that $$\cos{3x}+\cos{x}=2\cos{2x}\cos{x}$$ But I have no idea how these are equal.
A6SE
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How to find the value of $4\cos(\frac{\pi}{26})+\tan(\frac{2\pi}{13})$

I have found in wolfram alpha that $\displaystyle 4\cos\left(\frac{\pi}{26}\right)+\tan\left(\frac{2\pi}{13}\right)=\sqrt{13+2\sqrt{13}}$. How to prove this identity ? Thank you.
kong
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Why $\frac{1}{\cos(\sin^{-1}(x))}=\frac{1}{\sqrt{1-x^2}}$?

This is the inverse function of sin. Why is $\cos(\sin^{-1}x)=\sqrt{1-x^2}$? Thanks a lot.
Andrew
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