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.

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Did I get these trigonometric functions correct?

Ok so I came across another question. Once again it says let $\cot\theta= 4/3$, with $\cos\theta<0$. Find the remaining trigonometric functions. By using the identities I got: \begin{align} \sin\theta&=-3/5 \\ \cos\theta&=-4/5 \\ \tan\theta&=3/4 …
Sarah Rubenstein
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Dilation, shrink a triangle by $30\%$

I would like to know how I shrink a triangle by $30\%$ using dilation not changing its center?
Josh
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To prove $ \tan(A) + 2 \tan(2A) +4\tan4A + 8 \cot8A =\cot(A) $

To prove $ \tan(A) + 2 \tan(2A) +4\tan4A + 8 \cot8A =\cot(A) $. I tried to convert $\tan(4A)$ and $ \tan(8A)$ to $\tan(A$) and tried putting in L.H.S but it becomes a mess. Is there a easy and also intuitive way to do this? Thanks
Sophie Clad
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An algebraic expression for a trigonometric function

I am teaching myself math, and I have a question involving writing trigonometric expressions as algebraic expressions: Write $\cos(\tan^{-1}(u))$ as an algebraic expression. The "correct" answer, according to the student solutions guide that I am…
CarrotPatch
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$\cos(x-90)$ and $\cos(-(x-90))$ how can they be the same?

I plotted two lines in Desmos Calculator. $\cos(x-90)$ which looks exactly like a $\sin x$ graph. $\cos (-(x-90))$ which looks exactly the same. However, I thought that $f(-x)$ reflects the entire line in the $y$-axis. So why do the two lines…
vik1245
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Simplify Inverse Trigonometric Expression

The problem asks to simplify the expression $\arccos (\frac 3 5 \cos x + \frac 4 5 \sin x)$ where $x \in \; [\frac {-3\pi} 4 , \frac \pi 4]$. Here's my approach. Let $\frac 3 5 = r \cos y$ and $\frac 4 5 = r \sin y$. Therefore, $r^2 = 1 \implies r =…
Shinsekai no Kami
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Solve the equation $2\arcsin x=\arcsin(\frac{3}{4}x)$

$$2\arcsin x=\arcsin(\frac{3}{4}x)$$ so $x\in[-1,1]$ so we have: $2\arcsin x=y\Rightarrow\sin\frac{y}{2}=x$ and $\arcsin x=y \Rightarrow \sin y=\frac{3}{4}x\Rightarrow\frac{4}{3}\sin y=x$…
UfmdFkiF
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Solving trigonometric equations like $1-s\sin^2\theta=a\sin^6\theta+b\cos^6\theta$

For what values of $s\in\mathbb{R}$ does the following identity hold for all $\theta\in\mathbb{R}$: $$1-s\cos^2\theta\sin^2\theta=a\sin^6\theta+b\cos^6\theta\tag{1}$$ for some $a,b\in\mathbb{R}$? In other words, for what values of $s$ can the…
Anon
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What is the definition of the slope of a degenerate line segment in simple terms?

I have a (computer science) homework assignment which has this snippet: Treat the slope of a horizontal line segment as positive zero; treat the slope of a vertical line segment as positive infinity; treat the slope of a degenerate line segment…
Joshua Leung
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If $ \sin\theta + \cos\theta = \frac 1 2$, what does $\tan\theta + \cot\theta$ equal?

A SAT II question asks: If $ \sin\theta + \cos\theta = \dfrac 1 2$, what does $\tan\theta + \cot\theta$ equal? Which identity would I need to solve this?
Haim
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Solve trigonometric equation $ \cos x - x \sin x=0 $

I've been studying trigonometric functions lately and there's one problem I didn't manage to solve. It states that f is a function defined as following: $$ f : \mathbb R_+\to \mathbb R $$ $$ x \mapsto x\cos x $$ I'm supposed to find the…
Carw Lucas
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If $0< \alpha, \beta< \pi$ and $\cos\alpha + \cos\beta-\cos (\alpha + \beta) =3/2$ then prove $\alpha = \beta= \pi/3$

If $0< \alpha, \beta< \pi$ and $\cos\alpha + \cos\beta-\cos (\alpha + \beta) =3/2$ then prove $\alpha = \beta= \pi/3$ How do I solve for $\alpha$ and $\beta$ when only one equation is given? By simplification I came up with something like…
ray
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If $f(\sin x) = \cos2x$, find $f(\cos x)$

How do you even approach this question. Is there something fundamental that i'm missing here.
Julian Lopez
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If $ \sin \alpha = \frac 45 $ and $ \cos \beta = \frac{5}{13} $, prove that $ \cos \frac{\alpha-\beta}{2} = \frac{8}{\sqrt {65}} $

I can solve it easily if I assume that $ 0 < \alpha, \beta < \frac{\pi}{2}$ But there is no mention of the quadrants in which $ \alpha $ and $ \beta $ lie in. Is the question wrong ?
Nathuram
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Difficulty with proving a trigonometric relationship

I'm having a hard time proving the following: $$(\cot(x) - \csc(x))^2 = \frac{(\sec(x) - 1)}{(\sec(x) + 1)}$$ As far as I can tell this specific question has not been asked, but please let me know if this is a duplicate. I am trying to manipulate…
Benjamin Lowry
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