Tagged Questions
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
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 $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$.