Mathematics Stack Exchange
Mathematics Stack Exchange

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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Prove $\frac{\sin\theta}{1+\cos\theta} + \frac{1+\cos\theta}{\sin\theta} = \frac{2}{\sin\theta}$

How to prove: $\frac{\sin\theta}{1+\cos\theta} + \frac{1+\cos\theta}{\sin\theta} = \frac{2}{\sin\theta}$ Please help.
Arya
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Possible trig identity?

Is there a trigonometric identity for $\sin(ab)$? Thanks in advance! I can't find it anywhere. Bothering me a lot. For that matter, what about $\sin(a^{-1})$? Both of these for cosine, too, but if you can get any one of these 4, that would be…
louie mcconnell
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Find intersection of line segment in rectangle perimeter.

I want to find the best way to calculate the point in the perimeter of a rectangle in which a line segment intersects. p is a point inside the rectangle ($(0, 0)$ is in the center of the rectangle). $\theta$ is the angle of the of the line segment…
EmilioPelaez
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Proof this curious trigonometric identity

Proof that $$\cos^2{10^\circ} + \cos^2{50^\circ} - \sin{40^\circ}\sin{80^\circ} = \frac{3}{4}$$ I notice that $10^\circ + 80^\circ = 90^\circ$, and $50^\circ +40^\circ = 90^\circ$. I tried doing some manipulation but my efforts were futile. Any…
Yiyuan Lee
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Surface of earth seen is 1/4

At what height from the ground can one see exactly 1/4 th of the Earth's surface ? Take earth to be sphere of radius r.
user71408
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If $x+y+z=xyz$, find $\frac{3x-x^3}{1-3x^2}+\frac{3y-y^3}{1-3y^2}+\frac{3z-z^3}{1-3z^2}$

I found this question in a maths worksheet of trigonometry (kinda odd, right?), but I dont know how to figure it out. If $\displaystyle x+y+z=xyz$, find $\displaystyle\frac{3x-x^3}{1-3x^2}+\frac{3y-y^3}{1-3y^2}+\frac{3z-z^3}{1-3z^2}$ First I thought…
Rohinb97
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Trigonometric identity proof

I’ve got to proof that: $$\tan\left(\frac{A}{2}\right) = \sqrt{\frac{1 -\cos(A)}{1 + \cos(A)}}$$ My attempt was (tried with right side): $$= \pm \sqrt{\frac{1 -\cos(A)}{1 + \cos(A)} \cdot \frac{1 -\cos(A)}{1 - \cos(A)}}$$ $$= \pm \sqrt{\frac{\left(1…
Andrés Botero
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Does analytic closed form solution exist for this trigonometric equation?

solve for x: $\sin(ax)=k\sin(bx)$, a,b,k and x are real numbers I am looking for a very general solution when a,b and k are completely unrelated.
user130354
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Which is greater, $\cos(\cos(1))$ or $\cos(\cos(\cos(1)))$?

What is the simplest way we can find which one of $\cos(\cos(1))$ and $\cos(\cos(\cos(1)))$ [in radians] is greater without using a calculator [pen and paper approach]? I thought of using some inequality relating $\cos(x)$ and $x$, but do not know…
Sawarnik
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Using De Moivre's Theorem to prove $\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$ trig identity

I am stuck on trying to prove a trig identity using De Moivre's theorem. I have to prove, $$\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$$ I am not sure where to even start, I broke the LHS down to $$\cos(3\theta) + i\sin(3\theta)$$ but I have no…
user
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Trigonometric identity for $a\sin 2x + b\sin(x+\alpha )$

The following is a known trigonometric identity, $$a\sin x+b\sin(x+\alpha )=c\sin(x+\beta )\,$$ where $$c=\sqrt {a^2+b^2+2ab\cos \alpha },\,$$ and $$\beta =\arctan \left(\frac {b\sin \alpha }{a+b\cos \alpha }\right) + \begin{cases} 0 & \text{if }…
Heisenberg
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How do I determine the sign of $\sin\theta$ in this question.

$\alpha$ and $\beta$ are two different values of $\theta$ lying between $0$ and $2\pi$ which satisfy the equation $\begin{equation}6\cos\theta + 8\sin\theta = 9\end{equation}$. Find $\sin(\alpha+\beta).$ I solved and got $\cos(\alpha+\beta) =…
Shaurya Gupta
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How to remember trigonometric ratios for allied angles?

I just started studying trigonometry in unit circle and I want to know if there is some intuitive way to remember the value of $\sin(n\cdot\frac{\pi}{2}\pm\theta) \text{ and }\cos(n\cdot\frac{\pi}{2}\pm\theta)$, $n\in \mathbb N$. Sure I can use the…
Shaurya Gupta
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Showing $(1 + \sin x)^{-1} = \sec^2 x - \sec x \tan x$

I was asked to show the following: $$\frac {1}{1 + \sin x} = \sec^2 x - \sec x \tan x$$ I proceeded to factorize and simplify.. $\sec x (\sec x - \tan x)$$ $\sec x (\frac {1}{\cos x} - \frac {\sin x}{\cos x}) $ $\sec x (\frac {1 - \sin x}{\cos x})…
KillerKidz
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The value of $\cos^4\frac{\pi}{8} + \cos^4\frac{3\pi}{8}+\cos^4\frac{5\pi}{8}+\cos^4\frac{7\pi}{8}$

Problem : The value of $\cos^4\frac{\pi}{8} + \cos^4\frac{3\pi}{8}+\cos^4\frac{5\pi}{8}+\cos^4\frac{7\pi}{8}$ If this could have been like this $\cos\frac{\pi}{8} + \cos\frac{3\pi}{8}+\cos\frac{5\pi}{8}+\cos\frac{7\pi}{8}$ then we can take the…
Sachin
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