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
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Proving sine of sum identity for all angles

Could anyone present a proof of sine of sum identity for any pair of angles $a$, $b$? $$\sin(a+b) = \sin(a) \cos(b) + \cos(a) \sin(b)$$ Most proofs are based on geometric approach (angles are $<90$ in this case). But please note the formula is…
user4205580
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How to prove that $\left(\sqrt{3}\sec{\frac{\pi}{5}}+\tan{\frac{\pi}{30}}\right)\tan{\frac{2\pi}{15}}=1$

From this geometry problem, I can not find geometry solution. However the answer is $X=\frac{2\pi}{15}$ by geometry method. Then I get the identity $$\left(\sqrt{3}\sec{\frac{\pi}{5}}+\tan{\frac{\pi}{30}}\right)\tan{\frac{2\pi}{15}}=1.$$ How to…
kong
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Solving $\tan x-\tan(2x)=2\sqrt{3}$

$$\tan x-\tan(2x)=2\sqrt{3}$$ TRY #1 $$\begin{align*} \tan x-\tan(2x)=2\sqrt{3}&\implies\tan x=2\sqrt{3}+\tan{2x}\\ &\implies \tan^2x=\tan^2(2 x)+4 \sqrt{3} \tan(2 x)+12\\ &\implies\tan^2x=(\frac{2\tan x}{1-\tan^2 x})^2+4\sqrt{3}\frac{2\tan…
user205150
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Exact value of $\frac{\arccos(1-2\tan^2\alpha)}{2\arcsin(\tan\alpha)}$

Let $\alpha\in\left(0,\dfrac\pi2\right)$. What is the exact value of $$\dfrac{\arccos(1-2\tan^2\alpha)}{2\arcsin(\tan\alpha)}$$ Firstly, I tried to simplify $1-2\tan^2\alpha$ and got $$\dfrac{3\cos^2\alpha-2}{\cos^2\alpha}$$ What is next step? Is…
user164524
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Trigonometric equation, find $\sin \theta $

Find $\sin \theta $ if $a$ and $c$ are constants $$ 1-\left(c-a\tan\theta\right)^2=\frac{\sin^2\theta\cos^4\theta }{a^2-\cos^4\theta } $$
yogeesh
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Simplify a quick sum of sines

Simplify $\sin 2+\sin 4+\sin 6+\cdots+\sin 88$ I tried using the sum-to-product formulae, but it was messy, and I didn't know what else to do. Could I get a bit of help? Thanks.
rk_347
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Find the value of $\displaystyle\sqrt{3} \cdot \cot (20^{\circ}) - 4 \cdot \cos (20^{\circ}) $

How to find the value of $$\displaystyle\sqrt{3} \cdot \cot (20^{\circ}) - 4 \cdot \cos (20^{\circ})$$ manually ?
Quixotic
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Trigonometric identity expressed as a sum of fractions.

I am just trying to figure out why this happens: $$ \cos x + \frac{\sin ^2 x}{\cos x} = \frac{\cos^2x +\sin^2 x}{\cos x} $$ How do we get $\cos^2x$ in the numerator on the right-hand side? I just don't get it.
Jordan M
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Trigonometric equality $\cos x + \cos 3x - 1 - \cos 2x = 0$

In my text book I have this equation: \begin{equation} \cos x + \cos 3x - 1 - \cos 2x = 0 \end{equation} I tried to solve it for $x$, but I didn't succeed. This is what I tried: \begin{align} \cos x + \cos 3x - 1 - \cos 2x &= 0 \\ 2\cos 2x \cdot…
user21385
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$\sec\theta + \tan\theta=4$ find $\cos\theta$ given: $\theta\neq90$

I tried the following : \begin{align}\sec\theta + \tan\theta&=4\\ \frac1{\cos\theta} + \frac{\sin\theta}{\cos\theta}&=4\\ \frac{1+\sin\theta}{\cos\theta}&=4\\ \frac{1+\sin\theta}4&=\cos\theta\end{align} now don't know how to evaluate further ?
Arun Sharma
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Finding the Value of a Trigonometric Function

I am trying to solve a homework problem that has to do with deciding which of two trigonometric functions is greater. This would be simple to do with a calculator, but the instructions explicitly say not to, so I've hit a wall. For example, I have…
nmagerko
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how to solve $a\sin x+b\cos x$

Let's solve: $\sqrt{3}\sin x - \cos x=2$ The left hand side may be expressed as $R\sin(x+ \phi)$ We know that $R=\sqrt{3+1}=2$ We also know that $\tan \phi= \frac{-1}{\sqrt{3}}$ The solution to $\tan \phi=\frac{-1}{\sqrt{3}}$ has many solutions, for…
mathos
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Solving $2\arcsin\frac{x}{2}+\arcsin(x\sqrt{2})=\frac{\pi}{2}$

How do I solve this equation: $$ 2\arcsin\frac{x}{2}+\arcsin(x\sqrt{2})=\frac{\pi}{2} $$ We know that: $$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$$ So letting $\alpha = 2\arcsin\frac{x}{2}$ and $\beta=\arcsin(x\sqrt{2})$ leads to:…
hohner
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Proving $\frac{\sin x + \sin 2x + \sin3x}{\cos x + \cos 2x + \cos 3x} = \tan2x$

I need to prove: $$ \frac{\sin x + \sin 2x + \sin3x}{\cos x + \cos 2x + \cos 3x} = \tan2x $$ The sum and product formulae are relevant: $$ \sin(A + B) + \sin (A-B) = 2 \sin A \cos B \\ \sin(A + B) - \sin (A-B) = 2 \cos A \sin B \\ \cos(A + B) + \cos…
hohner
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Proving Identities.

I tried to solve it but I cant get the answer. How to prove this by using a hand? $$ \sec^2x + \csc^2x = \sec^2x \csc^2x $$ $$ \frac{\sec\theta + 1}{\sec\theta - 1} = \frac{1 + \cos\theta}{1 - \cos\theta}$$ $$ \frac{1 - \cot^2\theta}{1 +…
user2936034
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