All Questions
Tagged with logarithms trigonometry
215
questions
-4
votes
0
answers
108
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Find $\arcsin c$, $\, c\in\Bbb C$ [duplicate]
A math fan sent me a solution of the weird equation $\sin z=2$ posted in Quora. It is Weird because in real calculus, we experienced that $-1\leq \sin x\leq 1$. I saw this question in so many places ...
- 11.9k
2
votes
1
answer
78
views
Solve the equation: $ \log_{\sin x} (\cos x) - 2 \log_{\cos x} (\sin x) + 1 = 0. $ [closed]
Solve the equation:
$ \log_{\sin x} (\cos x) - 2 \log_{\cos x} (\sin x) + 1 = 0. $
Attempt: I transorm this equation in $(\log\cos x-\log\sin x)(\log\cos x+2\log\sin x)=0$, therefore $\cos x=\sin x$ ...
user1316790
4
votes
1
answer
68
views
Logarithmic trig equation. Why is my solution wrong?
This is the problem: $\log^2_{4}{\cos2x} = \log_{\frac{1}{16}}{\cos2x}$.
My solution:
$$\log^2_{4}{\cos2x} = -\frac{1}{2}\log_{4}{\cos2x}$$
$$\log_{4}{\cos2x}(\log_{4}{\cos2x} + \frac{1}{2}) = 0$$
$$\...
- 43
0
votes
2
answers
33
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Question regarding range of logarithmic function
The question regarding which i am asking this question is as follows
Find the domain and range of the function
$$f(x) = \log_e(\sin x)$$
I found the domain easily as for the above function $\sin(x) \...
- 356
3
votes
2
answers
78
views
Logarithmic Equation Involving Trigonometric Functions
Solve the following equation in real numbers:
$\log_2(\sin x) + \log_3(\tan x) = \log_4(\cos^2 x) + \log_5(\cot x)$
My approach:
$\log_2(\sin x) + \log_3\left(\frac{\sin x}{\cos x}\right) = \log_2(\...
- 637
0
votes
1
answer
124
views
Prove $\frac{1}{2}\ln(1+n)<\sin(\frac{1}{2})+\sin(\frac{1}{4})+\sin(\frac{1}{6})+ ...+\sin(\frac{1}{2n})<\frac{1}{2}\ln(n)+\ln(2)?$
Problem
Prove that $$\frac{1}{2}\ln(1+n) < \sin\left(\frac{1}{2}\right) + \sin\left(\frac{1}{4}\right) + \sin\left(\frac{1}{6}\right) + ... + \sin\left(\frac{1}{2n}\right) < \frac{1}{2}\ln(n) + \...
- 31
4
votes
2
answers
114
views
But when will $i^x=2$?
So I was looking through the homepage of Youtube to see if there were any math equations that I might be able to solve when I came across this video by Blackpenredpen asking if $i^x$ will ever equal $...
- 2,652
-2
votes
1
answer
337
views
How do you solve: $\log_{\cos(x)} \sin(x) + \log_{\sin(x)} \cos(x) \le 2$ [closed]
I have the following inequation:
$$\log_{\cos(x)} \sin(x) + \log_{\sin(x)} \cos(x) \le 2$$
I know that $\sin(x)$ and $\cos(x)$ will give values in the interval $[-1, 1 ]$ but in the base there can't ...
- 1
2
votes
1
answer
190
views
How do I compare $\ln(x)$ to $\ln(x - 1)$?
When I was playing around with the formula for $\sin(\theta)$, I found out that $\ln(2i\sin(\theta)) = \ln(e^{2i\theta}-1) - i\theta$. Using common sense, I could derive that this would equal a little ...
2
votes
2
answers
156
views
How to avoid "impossible" linears with trig integrals
Let's say I want to integrate $\int\sec^3xdx$. Due to the way this expression is set up, you must use integration by parts, and not u-sub, etc.
Applying integration by parts, I get $\sec{x}\tan{x}-\...
- 467
1
vote
0
answers
103
views
What happens to units of values treated with sin, cos...?
I know what happens when I compute a square root of 9 square metres - it's just 3 metres. But what about more weird cases, like:
sqrt(4m) - square root of 4 metres - is it 2 square-roots-of-metres of ...
- 11
0
votes
1
answer
78
views
Derive an equation of the form Y = MX + C from $y\:=\:px^2+q\sqrt{x}$, where p and q are constants
Hello and Good day to you all. I have been trying to linearize the following equation to the form Y = MX + C in order to plot a straight a line graph with a given set of x and y values. I have arrived ...
- 125
4
votes
0
answers
88
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Why is $\int_{0}^{2\pi} \int_0^{2\pi} \frac{\ln(21-4(\cos x+\cos y+\cos(x+y)))}{2\ln(9/2)}\frac{dx}{2\pi} \frac{dy}{2\pi}$ almost $1$?
Consider the function
$$ f(x,y) = \frac{\ln(21-4(\cos(x)+\cos(y)+\cos(x+y)))}{2\ln(9/2)} $$
Its average value is awfully close to unity:
$$ \int_{0}^{2\pi} \int_0^{2\pi} f(x,y) \frac{\mathrm dx}{2\pi} ...
- 1,436
1
vote
2
answers
144
views
Solving $\cos(\log_{4x}(x+1))-\cos(\log_{4x}(4-x))\lt\log_{4x}(4-x)-\log_{4x}(x+1)$
There is an interesting inequality I've stumbled upon on the Internet. It has logarithms and trigonometry, but in contrast to something like this that uses trigonometry and logarithms separately, this ...
- 846
4
votes
1
answer
116
views
Definite integral of the logarithm of a trigonometric polynomial
Let $p$ and $q$ be two real numbers such that $q>p^2$, so that $1+2px+qx^2>0$ for all $x$. I need to calculate the integral
\begin{equation*}
\int_0^\pi\ln\big(1+2p\cos\theta+q\cos^2\theta\big)d\...
- 73