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.

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Finding the range of $\frac{1+\sin^4 x}{\sin^4 x}\cdot \frac{1+\cos^4 x}{\cos^4 x}$

Range of function $$f(x) = \frac{(1+\sin^4 x)}{\sin^4 x}\cdot \frac{(1+\cos^4 x)}{\cos^4 x}$$ Using $\bf{A.\geq G.M}$ Inequality, $$1+\sin^4 x\geq 2\sin^2 x \qquad\text{and}\qquad 1+\cos^4 x\geq 2\cos^2 x$$ So $$\frac{(1+\sin^4 x)}{\sin^4 x}\cdot…
juantheron
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Determining linear equation for intersection of $y=\sin x$, $y=\cos x$

I'm trying to solve the following problem: The graphs for $y = \sin x$ and $y = \cos x$ has two points of intersection in the interval $[-\pi, \pi]$. Determine the equation for the line that passes through these two points. What I've done (and…
tereskopu
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Proving the trigonometric identity

Please help me in proving the following idenity: $$8\cdot \cos 40^\circ\cdot \cos 20^\circ \cdot \cos 10^\circ = \cot 10^\circ$$
user39471
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Number of Solutions to the equation $\sin x=mx.$

I would like to know if there any analytical methods that can be used to solve this equation. So far, I've made the following observations: We have to only check for solutions within the domain $-1/m\leq x\leq1/m.$ More than one solution is…
Student
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How to compute $\sin^{-1}(\cos 2x)$ in different domains?

What is $\sin^{-1}(\cos 2x)$ when $x \in [\pi/2,3\pi/2]$? I tried $$\sin^{-1}(\cos 2x)=\sin^{-1}(\sin (\pi/2-2x))=\pi/2-2x$$ However, it turns out to be the solution when $x \in [0, \pi/2]$. and the solution when $x \in [\pi/2,3\pi/2]$ is…
Matata
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Expressing in the form $A \sin(x + c)$

Express in the form $A\sin(x+c)$ a) $\sin x+\sqrt3\cos x$; b) $\sin x-\cos x$ sol: a) $A=\sqrt{1+3}=2$, $\tan c=\frac{\sqrt 3}1$, $c=\frac\pi3$. So $\sin x+\sqrt3\cos x=2\sin(x+\frac\pi3)$ b) $\sqrt 2\sin(x-\frac\pi4)$ Can someone please…
Py42
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Explain the concept behind solving $\sin(x)\cos(x) + \cos(x) = 0$, from Paul's Online Math Notes

$$\sin(x)\cos(x) + \cos(x) = 0$$ You are asked to find all possible solutions. What I immediately did was bring over the $\cos(x)$ term and then divided across by $\cos(x)$ and then proceeded from there: $$\sin(x)\cos(x) = -\cos(x)$$ $$\sin(x) =…
RonGiant
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Maximum value of trigonometric equation $\cos^2(\cos\theta) + \sin^2(\sin\theta)$

For any real $\theta$ the maximum value of $$\cos^2(\cos\theta) + \sin^2(\sin\theta)$$ A. $1$ B. $1 + \sin^21$ C. $1 + \cos^21$ D. does not exist I tried it by converting the whole expression into $\sin$ but getting nowhere with that.…
Heisenberg
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How to find the solutions to $\frac{11}{2} x - \cos x = 0$?

Find the values of $x$ such that $\frac{11}{2} x - \cos x = 0$. I really don't know how to find the solutions of this equation. I would appreciate if you could help me.
Oscar Gutierrez
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solve $\tan{x} = \tan{3x}$

I'm asked to solve $\tan{x} = \tan{3x}$ Here's my attempt: $$\tan{x} = \tan{3x}$$ $$\tan{x} = \tan{(x + 2x)}$$ $$\tan{x} = \frac{\tan{x} + \tan{2x}}{1-\tan{x}\tan{2x}}$$ Recall the identity: $$\tan{2x} = \frac{2\tan{x}}{1-\tan^2{x}}$$ So then we…
user360165
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Can we express the value of $\sin 1^\circ$ without using the imaginary unit?

I've been playing with sine of integer-degree angles; that is, $\sin\left(\frac{k \pi}{180}\right)$, where $k$ is an integer. I've noticed that you can divide the angle by $2$ and get sine of smaller and smaller angle by solving a quadratic equation…
user16320
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Problem in the solution of a trigonometric equation $\tan\theta + \tan 2\theta+\tan 3\theta=\tan\theta\tan2\theta\tan3\theta$

I needed to solve the following equation: $$\tan\theta + \tan 2\theta+\tan 3\theta=\tan\theta\tan2\theta\tan3\theta$$ Now, the steps that I followed were as follows. Transform the LHS first: $$\begin{split} \tan\theta + \tan 2\theta+\tan…
codetalker
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How to simply: $\prod_{k=1}^{100}(1+2cos\frac{2\pi.3^k}{3^{100} +1})$?

$$\prod_{k=1}^{100}\left(1+2\cos\left(\frac{2\pi.3^k}{3^{100} +1}\right)\right)$$ equals .I tried do this problem many time but i don't figure outhow to do this problem .
user108430
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I was trying to find out the intervals where $\sin ^{-1}x > \cos ^{-1}x$

I was trying to find out the intervals where $\sin ^{-1}x > \cos ^{-1}x$ The easiest way was to just look at the graph and I found out that the region is $x \in ({1\over \sqrt{2}} , 1]$ But I tried to prove the statement algebraically also but…
Harsh Sharma
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Solving $\cos^{-1}x+\cos^{-1}2x=-\pi$ gives the invalid answer $x=0$

To solve the inverse trigonometric equation $$\cos^{-1}x+\cos^{-1}2x=-\pi,$$ I use the normal cosine addition \begin{align} \cos^{-1}(2x^2 -\sqrt {1-x^2} \sqrt{1-4x^2})&=-\pi\\ 2x^2 -\sqrt {1-x^2} \sqrt{1-4x^2}&=\cos(-\pi)\\ 2x^2 -\sqrt {1-x^2}…
Harsh Sharma
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