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.
30,488 questions
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Let$\ \ f:[0,1]\to \mathbb{R}$ be a continuous function. Prove $\ \ \lim_{\lambda\to\infty}\int_0^1 f(x)\sin(\lambda x)\,dx = 0$.
Let$\ \ f:[0,1]\to \mathbb{R}$ be a continuous function. Show that $$\lim_{\lambda\to\infty}\int_0^1 f(x)\sin(\lambda x)\,dx = 0.$$
Here $f$ is only given to be continuous if $f$ was given to be ...
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Do there exist $\alpha, \beta \in (0, \pi/2)$ that are distinct rational multiples of $\pi$ such that $\frac{\sin \alpha}{\sin \beta}$ is rational?
The title says it all, so I'll provide the motivation here in the question body.
In Physics by Halliday, Resnick, and Krane, I encountered the following figure:
Assuming that the oscillating bodies ...
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Is the area of a cyclic polygon of nine or more sides greater than the square of the geometric mean of the sides?
The side lengths of a convex polygon do not uniquely determine the shape of the polygon but if the polygon is cyclic then the shape is uniquely determined by the side lengths. Consider a cyclic $n$-...
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Solve $\frac{\tan(a_1x)}{\tan(a_2x)}=\frac{a_1}{a_2}$ for $x$
I'm analyzing an electrical circuit described by a set of equations which I'm reporting here with as few parameters as possible:
$\begin{cases}
Z \tan(a_1 l)=a_1 \cdot k\\
Z\tan(a_2 l)=a_2 \cdot k
\...
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Simplify expression for lemniscate tangent function
$\def\sl{\operatorname{sl}}\def\cl{\operatorname{cl}}\def\tl{\operatorname{tl}}\def\cscl{\operatorname{cscl}}\def\secl{\operatorname{secl}}\def\cotl{\operatorname{cotl}}\def\d{\,\mathrm{d}}$
The ...
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Finding a function for $\sin(x)\sec(y) = \sin(y) + \sec(x)$
While messing around on desmos, I discovered the function $$\sin(x)\sec(y)=\sin(y)+\sec(x)$$ which appears as a warped sinusoid glide-reflected to fill the plane (graph in Desmos).
Each of these ...
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Golden ratio from Isosceles
I have the following Isososcles Triangle, and after some headscratching i managed to figure out the golden ratio. By Comparing the ratio of the longer segment to the shorter segment relative to the ...
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Find $\angle C$ given the relation $a^4 + b^4 + c^4 = 2a^2c^2+2b^2c^2$ [closed]
In a $\triangle ABC$ with sides $a, b, c$, the following relationship holds:
$$a^4 + b^4 + c^4 = 2a^2c^2+2b^2c^2$$
I need to determine the possible values for angle $C$.
My Attempt:
I suspect this ...
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Solving $3 \sin\theta + 4 \cos\theta = 4$ using the formula for $a \sin\theta + b \cos\theta$
I was recently trying to solve a trigonometry question, which asked to find theta:
$$3 \sin\theta + 4 \cos\theta = 4$$
I took $4 \cos\theta$ on the other side, and squared both the sides. After that I ...
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Show that $\frac{\cos(\alpha + \beta)}{\sin(\alpha - \beta)} = \frac{1}{3}$
Hello I am high school teacher a student gave me this question during private consultation
If $\cot(\alpha)\cot(\beta) = 2$ show that
$\frac{\cos(\alpha + \beta)}{\sin(\alpha - \beta)} = \frac{1}{3}$
...
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Evaluate (when possible) $\sum_{n=0}^{+\infty}\ln\left(2\cos\frac{\alpha}{2^n}-1\right)$
This appeared on the exercises sheet for a "Numerical Series" chapter of a university course: "Determine the nature and the possible sum of the numerical series". Among 18 examples ...
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predicting the point of changing trend types in tan nesting functions
(from a hsgs high school math group chat)
Let there be function $t(n,k)=q$, such that $k$ is in radians, and that $n$ is how many $\tan$ functions are nested in each other such that
$$t(1,k)=\tan(k)$$
...
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Is there an algebraic solution for all extrema of $f(x)=\sin(x)\sin(cx)$?
I am currently trying to find out, if there is an algebraic solution for all the extrema of $f(x)=\sin(x)\sin(cx)$?
Taking the derivative according to the product rule gives:
$f'(x)=\sin(x)\cos(cx)c + ...
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What is the probability distribution for the distance between two points placed randomly inside circles of different radii?
It is known that the probability density function for the distance, $s$, between two points located uniformly randomly inside a circle of radius $R$ is given by:
$$
f(s)=\frac{4s}{\pi R^{2}}cos^{-1}\...
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A really weird limit: $\lim_{n \to \infty, \, n \in \mathbb{N}} \frac{\tan(n^2+1)}{n!}$
I am trying to evaluate the following limit:
$$\lim_{n \to \infty} \frac{\tan(n^2+1)}{n!}, \qquad n \in \mathbb{N}.$$
It seems to me that this limit should be $0$, but I would like to understand how ...
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