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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If $x$, $y$, $z$ are in arithmetic progression, show that $\frac{\sin x + \sin y + \sin z }{\cos x + \cos y + \cos z} = \tan y. $

Show that if $x, y,$ and $z$ are consecutive terms of an arithmetic sequence, and $\tan y$ is defined, then $$\frac{\sin x + \sin y + \sin z }{\cos x + \cos y + \cos z} = \tan y. $$ I'm not sure what trig identities I would use and how to use them.…
rk_347
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Solving $2x - \sin 2x = \pi/2$ for $0 < x < \pi/2$

What is $x$ in closed form if $2x-\sin2x=\pi/2$, $x$ in the first quadrant?
Robert
  • 101
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Why is $(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000\cdots$?

Apparently, $$(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000000000000\cdots$$ Since it is a $2.0000000000\cdots$ instead of $2$, it isn't exactly $2$. Why is that?
Kurt
  • 47
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Verifying trig identities specific problem

$$\frac1{1-\cos y} + \frac1{1+\cos y} = 2\csc^2y $$ My attempt was me trying to find a common denominator on the left side but I don't know what to do after that.
Dhondup Tenzing
  • 51
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Show that $\frac{1-\cos2 \theta}{\sin2 \theta} = \tan \theta$

I have to show that the left equation simplifies to $\tan\theta$: Show that: $$\frac{1-\cos2 \theta}{\sin2 \theta} = \tan \theta$$ I do have prior knowledge that: $$\tan \theta = \frac{\sin\theta}{\cos \theta}$$ But I'm stuck from this point, I…
Hatmix5
  • 237
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What is the value of $\cos\left(\frac{2\pi}{7}\right)$?

What is the value of $\cos\left(\frac{2\pi}{7}\right)$ ? I don't know how to calculate it.
idm
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Rewrite the expression in the form $A \sin(x+C)$

Rewrite the following expression in the form $A \sin(x+C)$ $$4 \sin x + 4\sqrt{3} \cos x$$ This is what I have so far, and I'm not even sure it's the right approach. I just dont understand this concept as a whole: $$A \cos(c)\sin(x) +…
cocalope
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Constructively solving a trig equation

Solve the equation $$\frac{\sin(18°+x)}{\sin(x)}=\frac{\sin48°}{\sin18°}$$ If we use a computer we quickly note that $x=12°$, which can be easily proven: $$\sin18°=2\sin48°\sin12°=\cos36°-\cos60°$$ $$\iff…
chubakueno
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prove$\frac{ \sin a\vphantom{(}}{\sin b} +\frac{\cos a\vphantom{(}}{\cos b} = \frac{2\sin (a+b)}{\sin 2b}$

I've got this far but don't understand where the $2$ on the numerator comes from: $$\dfrac{\sin a \cos b + \cos a \sin b}{\sin b \cos b}\overset{?}{=}\dfrac{\sin(a+b)}{\sin 2b}$$
ejwm
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Using a Calculator with Decimal Degree Measures

So I have a problem with what I am being asked to do. Normally, when solving for the degree measures of trigonometric functions, I am presented with a fraction of rational numbers or a fraction with a rationalized radical. What I have now is a…
nmagerko
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Is it possible to expand $\sin(2x+1)\cdot\sin(2x+1)$?

Is it possible to treat it as a binomial?
Kapooky Handy
  • 595
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4 answers

Simplify: $\sin \frac{2\pi}{n} +\sin \frac{4\pi}{n} +\ldots +\sin \frac{2\pi(n-1)}{n}$.

Can you help me solve this problem? Simplify: $\sin \dfrac{2\pi}{n} +\sin \dfrac{4\pi}{n} +\ldots +\sin \dfrac{2\pi(n-1)}{n}$.
pegah
  • 21
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Express $\sin4\theta$ in terms of powers of $\sin\theta$ and $\cos\theta$

As far as I know $\sin4\theta$ = $4\sin\theta \cos\theta$, but I don't know if that's correct or what to do from there?
Daryn Wilkinson
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$-1.4\sin 3x - 0.2 \cos 3x$ in the form $R \sin (3x+\alpha)$ such that $R>0$ and $0<\alpha<2\pi$

Write $-1.4 \sin 3x - 0.2 \cos 3x$ in the form $R \sin (3x+\alpha)$ such that $R>0$ and $0<\alpha<2\pi$ I found $R= \sqrt{(-1.4)^2+(-0.2)^2}= \sqrt{2}$ And $\alpha= \arctan \frac{0.2}{1.4}= \arctan \frac {1}{7}$ Now the problem is I could write…
M.S.E
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Calculate $\sin(x)$, $\cos(x)$, and $\tan(x)$ without calculator

I know: $$\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}$$ $$\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ $$\tan(x) = \frac{\text{opposite}}{\text{adjacent}}$$ but how do you calculate $\sin(x)$, $\cos(x)$, and $\tan(x)$ without…
TheCoffeeCup
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