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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How can I prove that $\operatorname{arctg}(x) + \operatorname{arctg}(\frac{1}{x}) = \frac{\pi}{2}$, given that $x > 0$?

Which would be the easier way to prove that $\operatorname{arctg}(x) + \operatorname{arctg}(\frac{1}{x}) = \frac{\pi}{2}$ in cases where $x > 0$? I don't need explicit solutions, rather keywords and pointers towards the direction of a feasible…
András Hummer
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How to compute the sum $\cot^2\left(\frac{\pi}{9}\right)+\cot^2\left(\frac{2\pi}{9}\right)+\cot^2\left(\frac{4\pi}{9}\right)=~?$

How to compute the sum of $\cot^2\left(\frac{\pi}{9}\right)+\cot^2\left(\frac{2\pi}{9}\right)+\cot^2\left(\frac{4\pi}{9}\right)=~?$ The answer is $9$. I tried to use the formula $\cot (2\theta)=\dfrac{\cot^2\theta-1}{2\cot\theta}$ but it is getting…
Samar Imam Zaidi
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Range of Trigonometric function having square root

Finding range of function $\displaystyle f(x)=\cos(x)\sin(x)+\cos(x)\sqrt{\sin^2(x)+\sin^2(\alpha)}$ I have use Algebric inequality $\displaystyle -(a^2+b^2)\leq 2ab\leq (a^2+b^2)$ $\displaystyle (\cos^2(x)+\sin^2(x))\leq 2\cos(x)\sin(x)\leq…
Priti Bisht
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$ \text{sec}^2x + 3\text{cosec}^2x =8 $ Solving this trigonometric equation.

I have found two different methods of solving this trigonometric equation : $$ \text{sec}^2x + 3\text{cosec}^2x =8 $$ But these methods give different answers. Solution 1 $$ \text{sec}^2x + 3\text{cosec}^2x =8 $$ $$\implies \frac{1}{\text{cos}^2x}…
Regan Rogers
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If $\sin\theta+\sin\phi=a$ and $\cos\theta+ \cos\phi=b$, then find $\tan \frac{\theta-\phi}2$.

I'm trying to solve this problem: If $\sin\theta+\sin\phi=a$ and $\cos\theta+ \cos\phi=b$, then find $\tan \dfrac{\theta-\phi}2$. So seeing $\dfrac{\theta-\phi}2$ in the argument of the tangent function, I first thought of converting the left-hand…
Alraxite
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Evaluating $\cos(\alpha+\beta+\gamma)$

I am trying to evaluate $\cos(\alpha+\beta+\gamma)$ This is what I have done so far: I know $\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$ and $\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$ Treating…
mikoyan
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If $\alpha +\beta = \dfrac{\pi}{4}$ prove that $(1 + \tan\alpha)(1 + \tan\beta) = 2$

If $\alpha +\beta = \dfrac{\pi}{4}$ prove that $(1 + \tan\alpha)(1 + \tan\beta) = 2$ I have had a few ideas about this: If $\alpha +\beta = \dfrac{\pi}{4}$ then $\tan(\alpha +\beta) = \tan(\dfrac{\pi}{4}) = 1$ We also know that $\tan(\alpha…
mikoyan
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Transforming trigonometric identities

The problem goes like this: If $$N=2\sec^4x-3\sec^2x+2=\frac{\cos^2x}{\cos^2y}$$ Calculate the equivalent of $$M=2\tan^4x+3\tan^2x+2$$ The alternaties I have are: $$\frac{\tan^2x}{\tan^2y},\mbox{ }\frac{\tan^2y}{\tan^2x},\mbox{…
chubakueno
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Need help in proving that $\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$

We need to prove that $$\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$$ I have tried and it gets confusing.
Mahesh Natarajan
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Is it possible to calculate sine by hand?

Without a calculator, how can I calculate the sine of an angle, for example 32(without drawing a triangle)?
user80458
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Find the exact value of $\tan⁡(\cos^{-1} (-\sqrt{3}/2))$.

The question: Find the exact value of $\tan⁡(\cos^{-1} (-\sqrt{3}/2))$. The link is the image of the method I used. However it isn't the right answer, how come this method doesn't work?
sarah
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For which values of $\theta$ does $\sin(\theta)$ etc have closed form solutions?

There are special arguments to trigonometric functions which produce closed-form results, e.g. $\sin(\frac{\pi}{4}) = \frac{1}{\sqrt 2}$. I also recently learned that $\cos(\frac{\pi}{5})=\frac{\phi}{2}$ as shown here. We can extend these results,…
spraff
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How do I find the exact value of $\cos\frac{\pi}{12}\cos\frac{5\pi}{12}\cos\frac{7\pi}{12}\cos\frac{11\pi}{12}$?

I know that $\cos(6\phi)\equiv32c^6-48c^4+18c^2-1$ where $c=\cos\phi$. I also know that when $\cos(6\phi)=0$, then $\phi=\frac{k\pi}{12}$ ($k = 1,3,5,7,9,11$). How do I find the exact value of: $$\cos\left(\frac{\pi}{12}\right)…
maxmitch
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How to find $(2-\sec^2 1^\circ)(2-\sec^2 2^\circ)\cdots \overline{(2-\sec^2 45^\circ)}\cdots(2-\sec^2 89^\circ)$

Evaluate $$(2-\sec^2{1^{\circ}})(2-\sec^2{2^{\circ}})(2-\sec^2{3^{\circ}})\cdots(2-\sec^2{44^{\circ}})(2-\sec^2{46^{\circ}})\cdots(2-\sec^2{89^{\circ}})$$ This same problem come from Problem 21:But that problem is very very easy…
math110
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