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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how do you learn trigonometric identities

Possible Duplicate: Is there a more efficient method of trig mastery than rote memorization? i find myself loosing it in 1st semester calculus, mainly because people are using trigonometric identities i never heard of before. Are those usually…
rollover
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Proving $1+\cos a + \cos 2 a+ \cdots + \cos(n-1)a = 0$, when $a=2\pi/n$ and $n$ is odd

I am trying to prove that: $1+\cos a+\cos 2a+\cos3a+\cos4a=0$ where $a=\frac{2\pi}5$ (pentagon arrangement). Actually this is true for any $n>1$: $1+\cos a+\cos2a+\dots+\cos(n-1)a=0$ (polygon) where $a={2 \pi\over n}$. Easy to show for even $n$…
user539511
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Solving a trig equation $1+ \sin (x)=2 \cos(x)$?

How would I solve the following equation? $$ 1+ \sin (x)=2 \cos(x) $$ I am having difficult with it.
Fernando Martinez
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$p(\sin x) = \sin(2x)?$

a) Are there polynomials $ p (x) $ satisfying $ p(\sin x) = \sin (2x) \quad\quad \forall x \in \mathbb{R} $ ? b) An extension of this problem is: 1) If $n$ is even, then there does not exist a polynomial $P$ satisfying $P\left(\sin…
Mat15
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Assume that $\alpha, \beta, \gamma \in [0,\pi/2]$, and $\sin\alpha+\sin\gamma=\sin\beta$, $\cos\beta+\cos\gamma=\cos\alpha$. Find $\alpha-\beta$.

Assume that $\{\alpha, \beta, \gamma\} \subset \left[0,\frac{\pi}{2}\right]$, $\sin\alpha+\sin\gamma=\sin\beta$ and $\cos\beta+\cos\gamma=\cos\alpha$. Try to find a value of $\alpha-\beta$. Actually I have gotten that $\alpha+2\gamma+\beta=\pi$…
Intoks Liobein
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Show that the cosine of $\theta$ is $\frac{1 - t^2}{1 + t^2}$. Where am I going wrong?

I've recently picked up the book Mathematics and it's History by John Stillwell due to a recent curiosity in the history of math. I started doing one of the exercises in the book and got a little stumped (I'll admit, I'm quite rusty at math). I'm…
MrDiddly
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Examples of trigonometric substitutions for solving equations

I was working through a booklet of Olympiad-style problems when I came across a method which used the substitution $x = \cos \alpha$ to solve $x = \sqrt{2 + \sqrt{2-\sqrt{2+x}}}$. The solution works out nicely using the half angle formula. Are there…
wrb98
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Trig Identities : $\frac{\sin (4x)}{1-\cos(4x)} \frac{1-\cos(2x)}{\cos(2x)} = \tan(x)$

I want to prove that $$\frac{\sin (4x)}{1-\cos(4x)} \frac{1-\cos(2x)}{\cos(2x)} = \tan(x)$$ \begin{align} \text{Left hand side} : & = \sin(2x+2x)/(1-\cos(2x+2x)) \times ((1-\cos^2x+\sin^2x)/(\cos^2x-\sin^2x))\\ & =…
DavidSalib
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Is it true that $\sinh^{-1}\Big(\frac{z}{\sqrt{1-z^2}}\Big)=\tanh^{-1}(z)$?

How does one prove that it is?
Acid2
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Prove that $\frac{(\sin 20^\circ + \cos 20^\circ)^2}{\cos 40^\circ} = \cot 25^\circ$

So I'm trying to come up with an answer to this question for hours now. I don't know what I'm doing wrong and none of the calculators on the internet couldn't help so I figured I should ask people. What have I done so far: $\frac{(\sin 20^\circ +…
user4709059
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Proving $\cos^n(x) = \sum_{k=0}^{n} a_k \cos(kx)$

I'm being asked to prove that, for some coefficients $a_k$, $$ \cos^n(x) = \sum_{k=0}^{n} a_k \cos(kx) $$ I think I got pretty close, $$ \begin{align} \cos^n(x) & = { \left( \frac{e^{ix} + e^{-ix}}{2} \right) ^n} \\ & = \sum_{k=0}^{n} {n \choose k}…
galah92
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Given the length of 3 sides and 2 points, how do I find the third point of a right triangle

Prologue: I have very moderate knowledge of mathematics (highschool sophomore level). Any explanation needs to be broken down to chewable bits. I'm sorry if this inconveniences you guys. I have the following problem: How do I find out if a circle…
FinnTheHuman
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Find $\sin A$ and $\cos A$ if $\tan A+\sec A=4 $

How to find $\sin A$ and $\cos A$ if $$\tan A+\sec A=4 ?$$ I tried to find it by $\tan A=\dfrac{\sin A}{\cos A}$ and $\sec A=\dfrac{1}{\cos A}$, therefore $$\tan A+\sec A=\frac{\sin A+1}{\cos A}=4,$$ which implies $$\sin A+1=4\cos A.$$ Then what to…
Bala
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Help with solving trig identity problem

It's been 20 years since I did trig, and this one seems a little tricky. How would I solve $$ \tan^2(x) -2\tan(x)=1 $$ with steps?
MaLio
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How can I prove this trigonometric equation with squares of sines?

Here is the equation: $$\sin^2(a+b)+\sin^2(a-b)=1-\cos(2a)\cos(2b)$$ Following from comment help, $${\left(\sin a \cos b + \cos a \sin b\right)}^2 + {\left(\sin a \cos b - \cos a \sin b\right)}^2$$ $$=\sin^2 a \cos^2b + \cos^2 a \sin^2 b + \sin^2…
Fawad
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