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 to get $ \cot(\theta/2)$ from $ \frac {\sin \theta} {1 - \cos \theta} $?

According to wolfram alpha, $\dfrac {\sin \theta} {1 - \cos \theta} = \cot \left(\dfrac{\theta}{2} \right)$. But how would you get to $\cot \left(\dfrac{\theta}{2} \right)$ if you're given $\dfrac {\sin \theta} {1 - \cos \theta}$?
Cameron Martin
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How to prove $\cos \theta + \sin \theta =\sqrt{2} \cos\theta$.

I'm learning Trigonometry right now with myself. I'm stuck in a problem from sometimes (If $ \cos \theta - \sin \theta =\sqrt{2} \sin\theta $, proof that $ \cos \theta + \sin \theta =\sqrt{2} \cos\theta $) . I don't know what to do next. Please…
Yogesh Tripathi
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How to solve $\tan{a}=1/2$

What is the simplest way to solve: $\tan{a}=1/2$ I know from Mathematica that the answer is around 0.463648, but how can I achieve this result using only pencil and paper?
user303932
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Trigonometry - Roof Structure Angles

I've been working through a trigonometry book and have been stuck on the following question for a while now. The diagram shows a roof structure. PQRS is a horizontal rectangle. The faces ABRQ, ABSP, APQ and BRS all make an angle of $45$ degrees…
guest501051
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Prove that $\dfrac{1}{2}\sqrt{4\sin^2(36^{\circ})-1} = \cos(72^{\circ})$

Prove that $\dfrac{1}{2}\sqrt{4\sin^2(36^{\circ})-1} = \cos(72^{\circ})$ This question seemed pretty simple so I first started out by turning the left-hand side into terms of $\cos(x)$. We have $\dfrac{1}{2}\sqrt{4(1-\cos^2(36^{\circ}))-1} =…
Jacob Willis
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How to prove $(\sin x)^{\sin x}<(\cos x)^{\cos x}$?

How to prove that $$(\sin x)^{\sin x}<(\cos x)^{\cos x}$$ if x is from $]0, \pi/4[$?
Sierra Res
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Simple question on trigonometry identities of sec and tan

Please, I want to know different methods to prove following identity $$\frac{\tan \theta + \sec\theta - 1}{\tan\theta-\sec\theta + 1}=\frac{1+\sin\theta}{\cos\theta}$$
mnulb
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Is it just a coincidence or is it related to how values of sine calculated?

Actually, one of my math teacher told me about this. I want to know is there any relationship between this trick and their respective values?
user1444692
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Trigonometric identity: Sum of the squares of sine, cosine and tangent.

One has that the sum of the squares of the functions sine, cosine and tangent satisfy $$ \sin^2(x) + \cos^2(x) + \tan^2(x) = \frac{d}{dx} \tan (x)$$ Does this have any nice interpretation or meaning, or is it simply just a complicated way of writing…
Improve
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Finding the exact values of $\sin 4x - \sin 2x = 0$

So I've used the double angle formula to turn $$\sin 4x - \sin 2x = 0$$ $$2\sin2x\cos2x - \sin2x = 0$$ $$\sin2x(2\cos2x - 1) = 0$$ $$\sin2x = 0$$ $$2x = 0$$ $$x = 0$$ $$2\cos2x - 1 = 0$$ $$2\cos2x = 1$$ $$\cos2x = \frac{1}{2}$$ $$2x = 60$$ $$x =…
Kinson Chan
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Solve the equation: $ \sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos3x$

How do I solve this equation: $$ \sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos3x$$
mathlover
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Rewriting $\sin(\arccos(y))$ and $\arcsin(\cos(x))$

Prove the following identities: \begin{align} (a) && \sin(\arccos(y)) &= \sqrt{1-y^2}\\ (b) && \arcsin(\cos(x)) &= \frac{\pi}{2}-x \end{align} For (a) I am not sure how I would get a root out of any identity. For (b) I can transform the $\cos$ to…
NotSure
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solving Trigonometric equations: $\textrm{cos}(5x)\textrm{cos}(x)=\textrm{sin}(5x)\textrm{sin}(x)-0.5$

Question: Find solutions for $\textrm{cos}(5x)\textrm{cos}(x)=\textrm{sin}(5x)\textrm{sin}(x)-0.5$ I did $\textrm{cos}(6x)=-1/2$ using the subtraction formula for cos. I'm confused how to find the solutions now since there are 12. I thought you…
maria
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Trigonometric equation $\sin x+1=\cos x$

$$\sin x+1=\cos x,\quad x\in[-\pi,\pi]$$ How do you solve by squaring both sides? the solution is $x\in\{-\pi/2,0\}$ so the solutions $\pi$ and $-\pi$ are inadmissible, I do not understand how by subbing $-\pi$ back into both sides of the equations…
melanie
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Identity of $8\sin^2(t)\cos^2(t)$

I know this probably has a simple answer, but I am having trouble understanding the steps to find the identity for this problem. This is the answer I was provided: $$8\sin^2(x)\cos^2(x) = 2\sin^2(2x)$$ The closest Identity I can find…
acupofjose
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