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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Interesting results on Ramanujan's trigonometric identity

In this paper of generalized Ramanujan's trigonometric identity, the author showed that: Let $p$ be a prime number congruent to 1 modulo 6, and $g$ a primitive root modulo $p$, i.e. a generator of the group $\mathbb F_{p} ^ {\times} \cong C_{p-1}$.…
Nanhui Lee
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How to find the value of $\sin{\dfrac{\pi}{14}}+6\sin^2{\dfrac{\pi}{14}}-8\sin^4{\dfrac{\pi}{14}}$

Determine $$ \sin\left(\pi \over 14\right) + 6\sin^{2}\left(\pi \over 14\right) -8\sin^{4}\left(\pi \over 14\right) $$ My idea: Let $\displaystyle{\sin\left(\pi \over 14\right)} = x$.
math110
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trigonometry with alternative parametrizations of the circle

In trigonometry it is conventional when defining $\cos\theta$, $\sin\theta$, etc., to parametrize the circle by arc length $\theta$. Some trigonometric identities don't depend at all on which parametrization of the circle is used, e.g.…
Michael Hardy
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Proof for the formula of sum of arcsine functions $ \arcsin x + \arcsin y $

It is known that \begin{align} \arcsin x + \arcsin y =\begin{cases} \arcsin( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \\\quad\text{if } x^2+y^2 \le 1 &\text{or} &(x^2+y^2 > 1 &\text{and} &xy< 0);\\ \pi - \arcsin( x\sqrt{1-y^2} + y\sqrt{1-x^2})…
Parth Thakkar
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How to prove $\frac{1+\sin{6^\circ}+\cos{12^\circ}}{\cos{6^\circ}+\sin{12^\circ}}=\sqrt{3}$?

I found this interesting result. Show that $$\dfrac{1+\sin{6^\circ}+\cos{12^\circ}}{\cos{6^\circ}+\sin{12^\circ}}=\sqrt{3}.$$ See this Wolfram Alpha output. My attempt is very ugly. We know…
math110
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How do I prove that $\cos{\frac{2\pi}{7}}\notin\mathbb{Q}$?

How do I prove that $\cos{\frac{2\pi}{7}}\notin\mathbb{Q}$? Should I use some geometrical approach or apagoge?
xzhu
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Are there any natural occurrences of taking a trig function of a trig function?

In devising challenging exercises for my students, I am tempted to give them something like $\cos(3\sin(4))$, but then I get to wondering whether such a calculation would ever be encountered in practice. Since radians are dimensionless, as are…
Mike Jones
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Can we express $\sin 1^\circ$ in a real closed, not repetitive, radical forms?

Can we express $\sin 1^\circ$ in a real closed, not repetitive radical forms? Any radical forms mean you can use any roots but without constants $\pi$, $e$ or other trigonometry functions.
kiss my armpit
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For which angles we know the $\sin$ value algebraically (exact)?

For example: $\sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$ $\sin(18^\circ) = \frac{\sqrt{5}}{4} - \frac{1}{4}$ $\sin(30^\circ) = \frac{1}{2}$ $\sin(45^\circ) = \frac{1}{\sqrt{2}}$ $\sin(67 \frac{1}{2}^\circ) = \sqrt{ \frac{\sqrt{2}}{4}…
John Alexiou
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How can we show $\cos^6x+\sin^6x=1-3\sin^2x \cos^2x$?

How can we simplify $\cos^6x+\sin^6x$ to $1−3\sin^2x\cos^2x$? One reasonable approach seems to be using $\left(\cos^2x+\sin^2x\right)^3=1$, since it contains the terms $\cos^6x$ and $\sin^6x$. Another possibility would be replacing all occurrences…
Martin Sleziak
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Solving $\sin^{2015}x+\cos^{2015}x=\frac12$

Find all the roots of $$\sin^{2015}x+\cos^{2015}x=\frac12\tag{1}$$ I'm a high school student, and this is my homework. This's my try: Let $\displaystyle t=\tan \frac x2\Rightarrow \sin x=\frac{2t}{1+t^2}, \ \ \cos x=\frac{1-t^2}{1+t^2}$ We'll have…
mja
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calculate x,y positions in circle every n degrees

I am having trouble trying to work out how to calculate the $(x,y)$ point around a circle for a given distance from the circles center. Variables I do have are: constant distance/radius from center ($r$) the angle from $y$ origin I basically need a…
John Cogan
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Calculate width and height of a rectangle, given its diagonal and ratio

Well, I know, it's easy. We did it in class some time ago and I forgot it, I'm stupid because I can't figure it out: E.g. I have a 32" TV with 16:9 ratio and I want to know its width and height. I'd like to know the whole derivation so I can…
Erik
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Is it possible to rewrite $\sin(x) / \sin(y)$ in the form of $\sin(z)$?

I'm looking to get a particular answer in the form of $\sin(z)$, and I managed to reach an answer in the form $\sin(x)/\sin(y)$. I've checked on a calculator which has confirmed that they're the same number, but how can I convert the fraction into a…
Dalekcaan1963
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Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$

Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$ My approach : I used $\sin A +\sin B = 2\sin(A+B)/2\times\cos(A-B)/2 $ $\Rightarrow \sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7} =…
Sachin
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