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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Solve $\sec (x) + \tan (x) = 4$

$$\sec{x}+\tan{x}=4$$ Find $x$ for $0
Ray Jasson
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How to find $ \tan \left(\frac{x}{2}\right) $ knowing that $\cos \left(x\right)+\sin \left(x\right)=\frac{7}{5} $

Good evening to everyone. I don't know how to find $ \tan \left(\frac{x}{2}\right) $ knowing that $$\cos \left(x\right)+\sin \left(x\right)=\frac{7}{5} $$ and x$\in (0,\frac{\pi}{3})$ Here's what I've tried: $$\tan \left(\frac{x}{2}\right) =…
T4yl0r
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How to find $\tan x $ from $(a+1)\cos x + (a-1)\sin x=2a+1$?

How do I find $\tan x$ from this equation? $$(a+1)\cos x + (a-1)\sin x=2a+1$$ Thanks for any help!!
Soham
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Prove that $\cos\frac {2\pi}{7}+ \cos\frac {4\pi}{7}+ \cos\frac {8\pi}{7}=-\frac{1}{2}$

Prove that $$\cos\frac {2\pi}{7}+ \cos\frac {4\pi}{7}+ \cos\frac {8\pi}{7}=-\frac{1}{2}$$ My attempt…
pi-π
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solving trigonometry equation $20\cot\theta + 15\cot\theta\operatorname{cosec}\theta - 4\operatorname{cosec}\theta = 3(1 + \cot^2 \theta) ?$

how do you solve for $0 < \theta < 360$: $$20\cot\theta + 15\cot\theta\operatorname{cosec}\theta - 4\operatorname{cosec}\theta = 3(1 + \cot^2 \theta) ?$$ I tried turning the $(1 + \cot^2 \theta)$ part into $\operatorname{cosec}^2$ then dividing…
kjhg
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Argument of a composite trigonometric function

While dealing with trigonometric functions, we usually assume the argument to be in radians. The operator then returns a unitless output. Ie: the range of sin x for instance would be a set of certain ratios. By the same logic, the argument of sine…
Aditya
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How to simplify $\sin(x-y)\cos(y)+\cos(x-y)\sin(y)$

the question How to simplify $\sin(x-y)\cos(y)+\cos(x-y)\sin(y)$ my steps I tried to use trig identities on the $\sin(x-y)$ and $\cos(x-y)$ and tried to distribute the others in but it didn't work. Any ideas?
John Rawls
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Trigonometric Ratios for angles greater than 90 degrees and the Unit Circle

I am confused about the Unit Circle explanation for the trigonometric ratios for angles greater than 90 degrees. It seems that for the first (top right) quadrant, $\sin(\theta)$ is equivalent to the y-coordinate, because $\sin(\theta)$ = opposite /…
Llama.new
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Find $\sin \theta $ in the equation $8\sin\theta = 4 + \cos\theta$

Find $\sin\theta$ in the following trigonometric equation $8\sin\theta = 4 + \cos\theta$ My try -> $8\sin\theta = 4 + \cos\theta$ [Squaring Both the Sides] => $64\sin^{2}\theta = 16 + 8\cos\theta + \cos^{2}\theta$ => $64\sin^{2}\theta -…
Amritanshu
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Given $A+B+C=180^{\circ}$, find value of $\tan A\cdot\tan B+\tan B\cdot\tan C+\tan A\cdot \tan C-\sec A\cdot\sec B\cdot\sec C$

Given $A+B+C=180^{\circ}$, find value of $$\tan A\cdot\tan B+\tan B\cdot\tan C+\tan A\cdot \tan C-\sec A\cdot\sec B\cdot\sec C$$ I know about some basic conditional identities but don't know how to use them here.
Rohit Sinha
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Evaluate using complex numbers: $\prod^{n}_{k=1}\cos\left(\frac{k\pi}{m}\right)$, where $m=2n+1$

Evaluate using complex numbers: $$\prod^{n}_{k=1}\cos\left(\frac{k\pi}{m}\right)$$ where $m=2n+1$. $\bf{My\; Try::}$ Let $\displaystyle P = \prod^{n}_{k=1}\cos\left(\frac{k\pi}{m}\right).$ Now let $\displaystyle \cos \left(\frac{k\pi}{m}\right) =…
juantheron
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Trigonometry- why we need to relate to circles

I'm a trigonometry teaching assistant this semester and have a perhaps basic question about the motivation for using the circle in the study of trigonometry. I certainly understand Pythagorean Theorem and all that (I would hope so if I'm a teaching…
Erik
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4 answers

Converting $\cos\phi$ into $\frac{1−t^2}{1+t^2}$, given that $t = \tan\frac{\phi}{2}$

I have to figure out the working to convert $\cos\phi$ into $\dfrac{1−t^2}{1+t^2}$, given that $t = \tan\dfrac{\phi}{2}$. It would be amazing if someone could help I've been trying to do it for hours.
user2733843
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Proving $\frac {\sin x}{1-\sin x}-\frac {\sin x}{1+\sin x}\equiv 2\tan^2 x$

I need assistance with proving the following identity: $$\frac {\sin x}{1-\sin x}-\frac{\sin x}{1+\sin x} \equiv 2\tan^2 x$$ What I have done so far is expanded them: $$\frac {\sin x\;(1+\sin x)}{(1-\sin x)(1+\sin x)}-\frac {\sin x\;(1-\sin…
M. Anderson
  • 75
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If $m \tan(\theta - \pi/6) = n \tan(\theta + 2\pi/3)$ then find $\cos 2\theta$

If $$m \tan(\theta - \pi/6) = n \tan(\theta + 2\pi/3)$$ then find $\cos 2\theta$ in terms of $m$ and $n$. What is the correct method to solve this question?
Jamil Ahmed
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