0
votes
1answer
48 views

How to find a point on the tangent line whos length is 1?

im trying to figure out a formula to find the point(x,y) on a tangent line whos length is between 0 and 1 while it rotates around the unit circle uniformly, so the point would either be right on the ...
12
votes
2answers
115 views

Proving the inequality $\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin(1/k^2)}{\cos^2 (1/(k+1))}$

How am I supposed to prove this inequality? $$\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin\left(\frac{1}{k^2}\right)}{\cos^2 \left(\frac{1}{k+1}\right)}$$ Jordan inequality might be an option but led me ...
5
votes
1answer
72 views

Find the following integral: [duplicate]

Find $$\int \sqrt{\tan x}dx$$ My attempt: $$\text{Let}\ I=\int \sqrt{\tan(x)}dx$$ $$\text{Let}\ u=\tan(x), du=(1+\tan^{2}(x))dx$$ $$I=\int \frac{\sqrt{u}}{u^{2}+1}$$ $$\text{Let}\ v=\sqrt{u}, ...
5
votes
1answer
78 views

Interesting definite integral involving exp and trig

I'm trying to evaluate the following integrals: $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \cos(\phi) d\phi$$ $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \sin(\phi) d\phi$$ for which I want to find ...
0
votes
2answers
66 views

Whats the maximum value of $ y=6\cos\left( \frac {2\pi}{14} x\right)-2?$

Please show the correct way how to do this thanks. I got this.. y will be maximum when cos2pi/14x - 2 is maximum i. e. when 2pi/14x- 2 = 0 [ cos0 is maximum = 1 ] so or 2pi/14x = 2 or 14x = pi ...
0
votes
1answer
72 views

Prove this proprety of $f(x)$

I've asked this question before a long time ago, but I didn't get a complete answer. This is the link to the incomplete answer: Prove the following property of $f(x)$? Let ...
3
votes
4answers
84 views

Trigonometric Formula

I am stuck with the simple expression $$ \frac{\cos^2(\theta + \alpha)}{1 - \cos^2(\theta - \alpha)} = \text{const.} $$ where $\theta$ is a variable and $\alpha$ is the number satisfying $$ \alpha = ...
2
votes
1answer
29 views

Cone shaped related rates of change question

A container is in the shape of a cone of semi-vertical angle $30^\circ $, with it's vertex downwards. Liquid flows into the container at ${{\sqrt {3\pi } } \over 4}{\rm{ }}c{m^{^3}}/s$ At the ...
2
votes
3answers
87 views

Fractional Trigonometric Integrands

$$∫\frac{a\sin x+b\cos x+c}{d\sin x+e\cos x+f}dx$$ $$∫\frac{a\sin x+b\cos x}{c\sin x+d\cos x}dx$$ $$∫\frac{dx}{a\sin x+\cos x}$$ What are the relations between the numerator in the denominator, and ...
0
votes
3answers
55 views

Differentiate $y = {(x + 2)^3}{(1 - \sin 2x)^2}{(1 + \tan x)^3}$

I haven't got very far in attempting this: $\eqalign{ & y = {(x + 2)^3}{(1 - \sin 2x)^2}{(1 + \tan x)^3} \cr & y = {\left( {(x + 2)(1 + \tan x)} \right)^3}{(1 - \sin 2x)^2} \cr} $ I'm ...
9
votes
2answers
168 views

Inequality $\sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0$

Show the following inequality for any $x\in [0, \pi]$ and $n\in \mathbb{N}^*$, $$ \sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0. $$ I have this question a very long time ago from a book or magazine but I ...
0
votes
1answer
63 views

Time average of $\cos^2 x$ function [closed]

How do I find the time average of $\cos^2(3-wt)$ ?
0
votes
2answers
25 views

Turning points on $2\sin x - x$

I'm self teaching and doing a book exercise which asks: "Considering only positive values of x, locate the first two turning points on the curve $2\sin x - x$ and determine whether they are maximum or ...
4
votes
5answers
130 views

Integrate $\int_0^\pi{{x\sin x}\over{1+\cos^2x}}dx$.

Integrate $\displaystyle \int \limits_0^\pi{{x\sin x}\over{1+\cos^2x}}dx$. I tried substituting $t=\cos x$, and then integrate with integration by parts. It got all messy... Thanks in advance for any ...
2
votes
2answers
62 views

$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$

$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$.

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