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.

29665 questions
3
votes
2 answers

Prove $\sin(a) < a < \tan(a)$ when $0 < a < \pi/2$

It is easy to prove that $\sin(a) < \tan(a)$ when $0 < a < \pi/2$, but how can I prove that $\sin(a) < a < \tan(a)$ when $0 < a < \pi/2?$
Kevin
  • 365
  • 1
  • 10
3
votes
2 answers

If $\sin\theta-\cos\theta\leq\mu (\cos\theta + \sin\theta)$, then $\tan\theta \leq \frac{1+\mu}{1-\mu}$

In a text I'm reading, an implication is made: $$\sin\theta-\cos\theta\leq\mu (\cos\theta + \sin\theta) \quad\implies\quad \tan\theta \leq \frac{1+\mu}{1-\mu}$$ I tried using some trigonometric identities to reproduce this result, but it seems I'm…
Curl
  • 305
3
votes
3 answers

Finding the number of solutions to $\cos^4(2x)+2\sin^2(2x)=17(1+\sin 2x)^4$ for $x\in(0,2\pi)$

Number of solution of the equation $\cos^4(2x)+2\sin^2(2x)=17(1+\sin 2x)^4\; \forall $ $x\in(0,2\pi)$ what i try $\cos^4(2x)+2\sin^2 2x=17(1+\sin^2(2x)+2\sin 2x)^2$ $1+\sin^4 (2x)=17(1+\sin^4 2x+2\sin^2 2x+4\sin^24x+4\sin 2x(1+\sin^2…
jacky
  • 5,194
3
votes
1 answer

Solve $y=\frac{1}{3}[\sin x+[\sin x+[\sin x]]]$ & $[y+[y]]=2\cos x$

$[x]$ represents the greatest integer function $y=\frac{1}{3}[\sin x+[\sin x+[\sin x]]]$ & $[y+[y]]=2 \cos x$ Find the number of solution My approach is as follows $\sin x \in (\pi,2\pi)$ $y=\frac{1}{3}[\sin x+[\sin x+[\sin x]]]$ $y=-1$ $[-1+[-1]]=2…
Samar Imam Zaidi
  • 8,800
3
votes
1 answer

What is the range of $g(x)=\cos^{2n+1}(x)+\sin^{2n+1}(x)?$

We have that the range of $f(x)=\cos^{2n}(x)+\sin^{2n}(x),\; n\in \mathbb{N},\; n\geq 2,\;x\in \mathbb{R}$ is $$f(x)\in[2^{1-n},1]$$ since with $t=\cos^2(x)$ such that $0\le t\le 1$, then $$t^n+(1-t)^n.$$ The stationary points are the roots…
George
  • 750
3
votes
1 answer

Confused about "Solve $5\cos\theta = 3\cot\theta$"

I recently got this question only half correct: "Solve for values of $\theta$ the equation $5\cos\theta = 3\cot\theta$, in the interval $0 \leq \theta \leq 360$" My solution was: $$5 \cos\theta = 3 \cot\theta$$ $$\frac{\cos\theta}{\cot\theta} =…
Danny King
  • 1,953
3
votes
1 answer

Proving this trigonometry proves difficult

At first the question appear cheap to me, so I hesitated in solving it but when I finally tried to solve it. I did not arrive at the supposed answer Given that $\tan A/2 = \csc A - \sin A $ Show that $\tan (A/2)^2 = -2 ± \sqrt{5}$ Using the…
Michael Umande
  • 115
3
votes
2 answers

What property was used in this sine transformation?

I have this expression: $$ ψ(χ) = A\sin^3(\frac{πχ}{α}) $$ And somehow the book i read equalizes the previous equation to this one: $$ ψ(χ) = \frac{A}{4}[3\sin(\frac{πχ}{α}) - \sin(\frac{3πχ}{α})] $$ What trigonometric identity was used to make this…
user1584421
  • 217
3
votes
3 answers

Prove $\lceil z \rceil=z+\frac12-\frac{\tan^{-1}(\tan(\pi(z+0.5)))}{\pi}$ when $z$ is not an integer

How can I prove that $\lceil z \rceil=z+\dfrac12-\dfrac{\tan^{-1}(\tan(\pi(z+0.5)))}{\pi}$ for all non-integer real numbers $z$? Z cannot be an integer because then tan(pi*z + pi/2) would be undefined. I got this equation by messing around with…
Ekadh Singh - Reinstate Monica
  • 640
3
votes
0 answers

Why do some books define $\cot x$ as reciprocal of $\tan x$?

I believe $\cot x$ should be defined as $ \dfrac{\cos x}{\sin x}$ and not as $\dfrac{1}{\tan x}$ Because $\cot x$ and $\dfrac{1}{\tan x}$ aren't even the same function, they have different domains. So for instance we know $\cot (π/2) = 0$ but…
William
  • 4,893
3
votes
2 answers

Can $\frac{\csc \alpha +\cos \alpha}{\cos \alpha - \tan \alpha - \sec \alpha}$ be simplified?

I am trying to simplify the following but I cannot. $$ \frac{\csc \alpha +\cos \alpha}{\cos \alpha - \tan \alpha - \sec \alpha} $$ Can it be simplified? Edit My last result is $$ - \frac{\cos \alpha \left( 1 + \sin \alpha \cos \alpha\right)} {\sin^2…
Display Name
  • 2,715
3
votes
2 answers

What is $\cos\left(\frac{\arctan(x)}{3}\right)$ and $\sin\left(\frac{\arctan(x)}{3}\right)$?

I know that $$\cos(\dfrac{\pi}{3} - \arctan(x))= \dfrac{1}{2\sqrt{(1+x^2)}} + \dfrac{\sqrt{3}x}{2\sqrt{(1+x^2)}}$$ $\cos\left(\dfrac{\pi}{3} - \dfrac{\arctan(x)}{3}\right)$ = ? $\cos\left(\dfrac{\pi}{3} - \dfrac{\arctan(x)}{3}\right) =…
JackLametta
  • 169
3
votes
5 answers

Eliminate $\theta$ from $\lambda\cos2\theta=\cos(\theta + \alpha) \space$ and $\space \space\lambda \sin2\theta=2\sin(\theta + \alpha)$

Eliminate $\theta$ from $\lambda \cos2\theta=\cos(\theta + \alpha)$ and $\lambda\sin2\theta=2\sin(\theta + \alpha)$ My approach: Dividing the RHS and LHS of both equations by $\lambda$, then squaring and adding them, we…
ami_ba
  • 2,132
3
votes
2 answers

A trigonometry question from STEP examination

Show that if at least one of the four angles A ± B ± C is a multiple of π, then $$\sin^4A + \sin^4 B + \sin^4 C − 2 \sin^2 B \sin^2 C − 2 \sin^2 C \sin^2 A − 2 \sin^2 A \sin^2 B + 4 \sin^2 A \sin^2 B \sin^2 C = 0$$ I want to start with proving…
Kevin
  • 101
3
votes
3 answers

Memorizing the unit circle?

I know a quick google brings up plenty of resources on memorization techniques for the unit circle but I thought I would get the math stack exchange's opinion. What is the best way to memorize the radian angles and their corresponding points on…
Matt
  • 1,123
1 2 3
…
99 100