Tagged Questions

6
votes
3answers
106 views

How can I find all the solutions of $\sin^5x+\cos^3x=1$

Find all the solutions of $$\sin^5x+\cos^3x=1$$ Trial:$x=0$ is a solution of this equation. How can I find other solutions (if any). Please help.
3
votes
2answers
75 views

Please, help me to find where is a mistake in the solutions of the equation.

I have this equation and I will be very thankful to anyone who can provide me any help with the one discrepancy in my solution and the solution from the self-learning website: $$ \frac{1+\tan(x) + ...
0
votes
1answer
50 views

Is there an analytic solution to the following equation

I have the following general equation in $x$ $$a\cos(b - cx) - d\cos(e - fx) = 0$$ with constants $a,b,c,d,e,f$. Is there an algerbraic solution to this or only a numeric one?
0
votes
1answer
45 views

Perplexities on the Weierstrass substituition for $\phi=\pi$

Let us suppose that we have a system of equations including trigonometric expressions in $\phi$ and we want to bound the number of possible solutions. If I apply the Weierstrass substituition ...
0
votes
4answers
86 views

For $\sqrt[3]{-1+i}$, is $r$ (when put in polar form) $\sqrt[6]{2}$?

And when you put that into the nth root form... It becomes $2^{1/18}\cos\theta + 2^{1/18}\sin\theta$? $n$th root form given is: $\sqrt[n]r\cdot\cos(\theta+2\pi k)n$
4
votes
3answers
264 views

solution to equation $a \cdot \cos(\theta) - b \cdot \sin(\theta) = c$

Does the equation $$ a \cdot \cos(\theta) - b \cdot \sin(\theta) = c$$ have a closed-form solution for $\theta$? What about the case where $a^2 + b^2 = 1$?
4
votes
2answers
182 views

Why isn't this square root $+$ or $-$?

I was tasked with proving the identity $\tan(\frac x 2) = \dfrac {\sin(x)}{1+\cos(x)}$ I used the quotient identity for tangent and the half angle identities for sine and cosine to get $ \pm \dfrac ...
0
votes
1answer
53 views

Polynomials with roots having the same module and linear dependent arguments

Is it possible for a polynomial with integer coefficients to have some of its roots: $$m_1e^{i\theta_1 \pi}, m_2e^{i\theta_2 \pi}, \ldots, m_ke^{i\theta_k \pi}$$ such that there exist nonzero integers ...
2
votes
0answers
67 views

References for “closed form” numeric solutions of $\tan x=-a x$

I am looking for references that discuss solutions of the equation $\tan x=-a x$ (for $x,a\in \mathbb{R}$). I know about the graphical approaches, and any number of numerical solution approaches, ...
10
votes
6answers
463 views

Complex roots of $z^3 + \bar{z} = 0$

I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre. Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
2
votes
1answer
116 views

Analytical method for root finding

Is there an analytical method to find the roots of the following equation? $$y = -\frac{1}{2}{x}^{2}-\cos(x)+1.1$$ I'm sorry for the trivial question, I'm new at math! :)
5
votes
2answers
301 views

How was Euler able to create an infinite product for sinc by using its roots?

In the Wikipedia page for the Basel problem, it says that Euler, in his proof, found that $$\begin{align*} \frac{\sin(x)}{x} &= \left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 ...
1
vote
2answers
79 views

Showing $2x=\left( 2n+1\right) \pi \left( 1-\cos x\right) $ has $2n+3$ roots when $n\in \mathbb{Z}_+$

I am struggling to show that the equation $$2x=\left( 2n+1\right) \pi \left( 1-\cos x\right) $$ where n is a positive integer, has $2n+3$ roots and no more and also if it possible to indicate their ...
4
votes
0answers
244 views

Find roots of sum of sinusoids

Given this function and an initial point, find the next root: $$ \begin{align} f(t) & = -L\\ & {} + A \sin(\Theta_1 + \omega_1 t) \\ & {} +B \cos(\Theta_1 + \omega_1 t)\\ & {} - ...
2
votes
3answers
506 views

Newton's method and trig functions on a computer

I'm trying to use Newton's method to find roots for the function $A \cos(\Theta_2 - \Theta_1) + B \sin(\Theta_1)$. (That is, iterate $x_{i+1} = x_i - f(x_i) / f'(x_i)$). I've got a working ...
5
votes
3answers
2k views

How to find roots of $X^5 - 1$?

How to find roots of $X^5 - 1$? (Or any polynomial of that form where $X$ has an odd power.)