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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Given that $\cos A + \cos B + \cos C = 0$ and $\sin A + \sin B + \sin C = 0$.

If $\cos A + \cos B + \cos C = 0$ and $\sin A + \sin B + \sin C = 0$. The value of $ \sin^3A+\sin^3B+\sin^3C$ What I can see here is that as $\sin A + \sin B + \sin C = 0$ hence $ \sin^3A+\sin^3B+\sin^3C=3\sin A \sin B\sin C$ but I am not able to…
Ananya
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Prove that $6(\sin^{10}A+\cos^{10}A)-15(\sin^8A+\cos^8A)+10(\sin^6A+\cos^6A)-1=0​$

Prove that ​$$6(\sin^{10}A+\cos^{10}A)-15(\sin^8A+\cos^8A)+10(\sin^6A+\cos^6A)-1=0​$$ Expression can be verified for different values of $A$ such as $\frac\pi4,\frac\pi2$ etc. But to prove it for general value of A?
H.P. Das
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Evaluate: $\csc^2\left(\frac{\pi}{9}\right)+\csc^2\left(\frac{2\pi}{9}\right)+\csc^2\left(\frac{4\pi}{9}\right)$

$$\csc^2\left(\frac{\pi}{9}\right)+\csc^2\left(\frac{2\pi}{9}\right)+\csc^2\left(\frac{4\pi}{9}\right) \;=\; \text{???}$$ $\bf{My\; Try::}$ Let $\displaystyle \frac{\pi}{9} = \theta\;,$ Then $9\theta = \pi\Rightarrow 6\theta = \pi-3\theta$ So…
juantheron
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Multiple Angle formulas, alternate forms

Relatively simple question, that might not be simple to answer: I have noticed that there are ways of expressing every double angle formula of a given trigonometric function using only that function except for $\sin$ and $\csc$. That…
Justin Benfield
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value of $\tan(A)$

What is value of $tanA$ if $2\tan(2A)+4\tan(4A)+\frac{8}{\tan(8A)}=0$ writing everything in $\tan(A)$ and solving for $t$ is next to impossible without maths engines. So i am seeking for a shorter way. Thanks
Archis Welankar
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Why is the double angle formula not used in this solution?

solve $\sin 2x = 0.5$ I am looking at the answer to this question and it does not use the double angle formula to make this: $\sin 2x = 2\sin x \cos x$ And instead use $2x = \arcsin 0.5$ Why is the double angle formula not used in this situation?
dagda1
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Prove $\frac{2\sec\theta +3\tan\theta+5\sin\theta-7\cos\theta+5}{2\tan\theta +3\sec\theta+5\cos\theta+7\sin\theta+8}=\frac{1-\cos\theta}{\sin\theta}$

Prove that $\frac{2\sec\theta +3\tan\theta+5\sin\theta-7\cos\theta+5}{2\tan\theta +3\sec\theta+5\cos\theta+7\sin\theta+8}=\frac{1-\cos\theta}{\sin\theta}$ $$\frac{2\sec\theta +3\tan\theta+5\sin\theta-7\cos\theta+5}{2\tan\theta…
Vinod Kumar Punia
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Show that the triangle is equilateral triangle

If a triangle $ABC$ has the equality $$h_a\cdot\sqrt{3} +\frac{a}{2}= b + c$$ $h_a$ is the height from $A$, then show that the triangle is equilateral. Using sine rule, I tried to show that bring equality to a form of showing that…
medicu
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Why does the graph of $\frac{1-cos(3x)}{3x}$ look so weird at small values

So I was just doing some homework, and I happened to graph this $\frac{1-cos(3x)}{3x}$ For some unknown reason I decided to zoom into to ridiculous levels and I saw a really really weird behavior of the graph. Sorta like a cartoony signal wave or…
J.Doe
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Trigonometric equation inversion

I am trying to invert the following equation to have it with $\theta$ as the subject: $y = \cos \theta \sin \theta - \cos \theta -\sin \theta$ I tried both standard trig as well as trying to reformulate it as a differential equation (albeit I might…
maruko
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find exact value of $\sin(10^\circ)$

How we can find exact value of $\sin(10^\circ)$? I tried trigonometric ways but I get this equation: $\ 8y^3-6y+1=0$ and $y = \sin(10^\circ)$ and all the roots are complex. I saw the pages in site, but I can't find the solution. Thanks
S.H.W
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Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$

Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$ $$\tan x+\sec x=2\cos x$$ $$\left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)=2\cos x$$ $$\left(\dfrac{\sin x+1}{\cos x}\right)=2\cos x$$ $$\sin…
Austin Broussard
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Why is the sine/cosine of an angle equal to its supplement?

I just got my hands on a trig text and I've been studying the law of sines and cosines so I can solve triangles other than right triangles. Something I've found odd while studying proofs of these theorems are the statements that the sine/cosine of…
user326604
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Simplify $\frac{1-\cos^2(-\theta)}{1+\tan^2(-\theta)}$

I'm doing practice problems out of Trigonometry 10th ed. Hornsby and ran into a question. Section 5.1 question 71: Write each expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression and all…
giraffesyo
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Sum of Dirichlet Kernels

I need to sum up several Dirichlet kernels. To do this, I would like to have a compact formula for $$ \sum_{n=-N}^N \sin(2nx+\xi) $$ where $x,\xi \in \mathbb R$. The final result should look like something similar to a product of two Dirichlet…
user42761
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