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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How does a calculator calculate the sine, cosine, tangent using just a number?

Sine $\theta$ = opposite/hypotenuse Cosine $\theta$ = adjacent/hypotenuse Tangent $\theta$ = opposite/adjacent In order to calculate the sine or the cosine or the tangent I need to know $3$ sides of a right triangle. $2$ for each corresponding…
themhz
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Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$?

Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?
Clayton Kershaw
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Are there theoretical applications of trigonometry?

I am a high school student currently taking pre-calculus. We have just finished a unit on analytic trigonometry. Are any purely theoretical uses for trigonometry? More specifically, can trigonometric concepts (or even functions) be used to…
Conan G.
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Is $\pi$ equal to $180^\circ$?

$$ \begin{array}{ccc} \sin{(\theta+180^{\circ})}=-\sin{\theta} & \cos{(\theta+180^{\circ})}=-\cos{\theta} & \tan{(\theta+180^{\circ})}=\tan{\theta} \\ \sin{(\theta+\pi)}=-\sin{\theta} & \cos{(\theta+\pi)}=-\cos{\theta} &…
Chin Huan
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Hidden patterns in $\sin(a x^2)$

I discovered unexpected patterns in the plot of the function $$f(x) = \sin(a\ x^2)$$ with $a = \pi/b$, $b=50000$ and integer arguments $x$ ranging from $0$ to $100000$. It's easy to understand that there is some sort of local symmetry in the plot…
Hans-Peter Stricker
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Are there any "nonstandard" special angles for which trig functions yield radical expressions?

Everyone learns about the two "special" right triangles at some point in their math education—the $45-45-90$ and $30-60-90$ triangles—for which we can calculate exact trig function outputs. But are there others? To be specific, are there any values…
WillG
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Prove that $\sum\limits_{k=1}^{n-1}\tan^{2}\frac{k \pi}{2n} = \frac{(n-1)(2n-1)}{3}$

How can you prove that $$ \sum_{k=1}^{n-1}\tan^{2}\left(\frac{k \pi}{2n}\right) = \frac{\left(n-1\right)\left(2n - 1\right)}{3} $$ for every integer $n\geq 1$ > ?. PS: no, it's not a homework... :-)
botismarius
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Prove $\sin^2\theta + \cos^2\theta = 1$

How do you prove the following trigonometric identity: $$ \sin^2\theta+\cos^2\theta=1$$ I'm curious to know of the different ways of proving this depending on different characterizations of sine and cosine.
Nick
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Is there an interpretation for this trigonometric identity?

A while ago I came across the following identity in an online math forum (of which I don't remember the name): $$\tan\left(\frac{\pi}{11}\right)+4\sin\left(\frac{3\pi}{11}\right)=\sqrt{11}.$$ It is not hard to give a proof by rewriting everything in…
carinii
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A question about the arctangent addition formula.

In the arctangent formula, we have that: $$\arctan{u}+\arctan{v}=\arctan\left(\frac{u+v}{1-uv}\right)$$ however, only for $uv<1$. My question is: where does this condition come from? The situation is obvious for $uv=1$, but why the inequality? One…
Johnny Westerling
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Where are the values of the sine function coming from?

On high school, I was taught that I could obtain any sine value with some basic arithmetic on the values of the following image: But I never really understood where these values where coming from, some days ago I started to explore it but I…
Red Banana
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Calculate $\frac{1}{\sin(x)} +\frac{1}{\cos(x)}$ if $\sin(x)+\cos(x)=\frac{7}{5}$

If \begin{equation} \sin(x) + \cos(x) = \frac{7}{5}, \end{equation} then what's the value of \begin{equation} \frac{1}{\sin(x)} + \frac{1}{\cos(x)}\text{?} \end{equation} Meaning the value of $\sin(x)$, $\cos(x)$ (the denominator) without using…
user2161721
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How were the sine, cosine and tangent tables originally calculated?

As I understand it... ahem... the (cosine, sine) vector was calculated for (30 degrees, PI/6), (45 degrees, PI/4) and (60 degrees, PI/3) angles etcetera, however, I would like know the original geometrical process for calculating the magnitudes for…
Dave Kirkby
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Disproving an "almost true" trigonometric identity

The plausible looking "identity" $$\sin(\frac{\pi}{51})+\cos(\frac{\pi}{74})=\frac{3}{2\sqrt 2}$$ is not true, but it is close indeed: $$LHS=1.0606\color{blue}{598...}$$ $$RHS=1.0606\color{red}{601...}$$ In fact the difference, on the order of…
user139000
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Proving $2 ( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} )$ is a root of$ \sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$

How can one show that the number $2 \left( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} \right)$ is a root of the equation $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$?
Amir Parvardi
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