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.

29665 questions
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Simplify trigonometric expression using trigonometric identities

I have the trigonometric expression: $$2\sin x +2\sin \left(\frac{\pi} {3} -x\right) $$ and it should simplified in: $$\sin x + \sqrt 3 \cos x$$ but I do not know what formulas to apply. Could you tell me how to simplify it?
zaz
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Integers $n$ satsifying $\frac{1}{\sin \frac{3\pi}{n}}=\frac{1}{\sin \frac{5\pi}{n}}$

If $\displaystyle \frac{1}{\sin \frac{3\pi}{n}}=\frac{1}{\sin \frac{5\pi}{n}},n\in \mathbb{Z}$, then number of $n$ satisfies given equation ,is What I tried: Let $\displaystyle \frac{\pi}{n}=x$ and equation is $\sin 5x=\sin 3x$ $\displaystyle \sin…
jacky
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Solve tan(x)+cos(x)=1/2

Is it possible (not numerically) to find the $x$ such as: $$ tan(x)+cos(x)=1/2 $$ ? All my tries finishes in a 4 degree polynomial. By example, calling c =…
pasaba por aqui
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How to find the roots of $\frac{\sqrt{3}-1}{\sin x}+\frac{\sqrt{3}+1}{\cos x}=4\sqrt{2}$?

Find all $x$ in the interval $(0,\pi/2)$ such that $\frac{\sqrt{3}-1}{\sin x}+\frac{\sqrt{3}+1}{\cos x}=4\sqrt{2}$. The options are (i)$\pi/9,2\pi/7$, (ii)$\pi/36,11\pi/12$ (iii)$\pi/12,11\pi/36$ (iv) All I have been able to find one value of $x$,…
MrAP
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Show an identity involving a sum of arctangents of algebraic expressions

Show that $$ 2\tan^{-1}\frac{\sqrt{x^2+a^2} - x + b}{\sqrt{a^2-b^2}} + \tan^{-1}\frac{x\sqrt{a^2-b^2}}{b\sqrt{x^2+a^2} + a^2} + \tan^{-1}\frac{\sqrt{a^2-b^2}}{b} = n\pi . $$ I tried using $$ x= a \tan \theta ,\; b= a \sin\phi,$$ but then…
maveric
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Show that $\cot{142\frac{1}{2}^\circ} = \sqrt2 + \sqrt3 - 2 - \sqrt6$.

Show that $\cot{142\frac{1}{2} ^\circ} = \sqrt2 + \sqrt3 - 2 - \sqrt6$. What I have tried: Let $\theta = 142\frac{1}{2}^\circ \text{ and } 2\theta = 285^\circ$. $$\cos 285^\circ = \cos 75^\circ$$ $$\cos 75^\circ = \frac{\sqrt3 -…
user642405
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Solving $\sin (100^\circ-x) \sin 20^\circ =\sin (80^\circ-x)\sin 80^\circ$

Solve for $x$ such that $$\sin (100^\circ-x) \sin 20^\circ =\sin (80^\circ-x)\sin 80^\circ$$ First, I use the co-function formula: $$\sin 80^\circ = \cos 10^\circ \tag{1}$$ Also, $$\sin 20^\circ = 2\sin 10^\circ \cos 10^\circ \tag{2}$$ From these,…
Heart
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Precalc Trig Identity, verify: $1 + \cos(x) + \cos(2x) = \frac 12 + \frac{\sin(5x/2)}{2\sin(x/2)}$

Working with LHS: I've tried using the sum to product trig ID to get: $1 + 2\cos(3x/2)\cos(x/2)$ from here I've tried a couple of things, but can't seem to get closer. I've tried changing the $(3x/2)$ into $(5x/2 - x)$ and using sum identity, but…
McMath
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Using the R method for finding all solutions to $\sin(2a) - \cos(2a) = \frac{\sqrt{6}}{2}$. My solution differs from official answer.

How many solutions does $$\sin(2a) - \cos(2a) = \frac{\sqrt{6}}{2}$$ have between $-90^\circ$ and $90^\circ$? I used the R method and got $$2a-45^\circ = \arcsin\left(\frac{\sqrt{3}}{2}\right).$$ Since $a$ is between $-90^\circ$ and…
SuperMage1
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Simplification of $\sqrt{(1-\cos\alpha \cos\beta)^2-\sin^2\alpha \sin^2\beta}$

Simplify the expression $$\sqrt{(1-\cos\alpha \cos\beta)^2-\sin^2\alpha \sin^2\beta}$$ I have done this way : $(1-\cos\alpha \cos\beta)^2 = 1-2\cos\alpha \cos\beta +\cos^2\alpha \cos^2\beta$ Please guide further....
Sachin Sharmaa
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How to prove $\frac{\tan (A)}{\tan (A)}+\frac{\cot (A)}{\cot (A)}=\frac{1}{1-2\cos(A)^2}$

I am unable to prove this trigonometric identity $$\frac{\tan (A)}{\tan (A)}+\frac{\cot (A)}{\cot (A)}=\frac{1}{1-2\cos^2(A)}$$ I have tried to transform the left-hand side and stuck with this $$\frac{2\sin(A)\cos(A)}{\sin(A)\cos(A)}$$ And I have…
user569622
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$X=(1 + \tan 1^{\circ})(1 + \tan 2^{\circ})(1 + \tan 3^{\circ})\ldots(1 + \tan {45}^{\circ})$. what is the value of X?

$$X=(1 + \tan 1^{\circ})(1 + \tan 2^{\circ})(1 + \tan 3^{\circ})\ldots(1 + \tan {45}^{\circ})$$ $$\tan(90-\theta)=\cot\theta=\frac{1}{\tan\theta}$$
HOLYBIBLETHE
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Number of solutions of the equation $\cos(\pi\sqrt{x-4})\cos(\pi\sqrt{x})=1$

Find the number of solutions of the equation…
Sooraj S
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Find a new point on a rectangle given an angle from the center

I am a software engineer and I'm trying to edit some images. Trying to find a good formula to find a new point on the rectangle, I know the center x and y coordinates, the height and width of the rectangle, and the theta in the direction of the…
Justin
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Solve $\sqrt{2}\sec x+\tan x=1$

Solve $\sqrt{2}\sec x+\tan x=1$ I understand it can be very easily solved by expanding in terms of $\sin x$ and $\cos x$, gives $x=2n\pi-\frac{\pi}{4}$. But, what if I do the following: $$ \sqrt{2}\sec x+\tan…
Sooraj S
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