All Questions
Tagged with cubics trigonometry
50
questions
1
vote
1
answer
108
views
Solving a cubic using triple angle for cos (i.e $\cos(3A)$)
a) Show that $x=2\sqrt{2}\cos(A)$ satisfies the cubic equation $x^3 - 6x = -2$ provided that $\cos(3A)$ = $\frac{-1}{2\sqrt{2}}$
I did not have a difficulty with this question, I have provided it for ...
- 15
2
votes
1
answer
104
views
Are there any ways to convert inverse trigonometric values to radicals?
When we solve a cubic equation $ax^3+bx^2+cx+d=0$, the roots are supposed to be in the form of radicals in real numbers or complex realm. However, if the discriminant is less than 0, the solution is ...
- 129
2
votes
3
answers
94
views
Stuck on simplifying expressions involving trig and inverse trig functions
TL; DR
Using Mathcad and Wolfram I can see that
$$\sqrt{7}\cos\frac{\tan^{-1}\left(\frac{9\sqrt{3}}{10}\right)}{3}=2.5$$
The decimal value seems to be exact because Mathcad displays it like that with ...
- 173
1
vote
1
answer
116
views
Solving $x^3-7x^2+14x-8-\frac12\sin x=0$
With the given problem:
$$x^3-7x^2+14x-8-\frac12\sin x=0$$
I factorized the cubic part:
$$x^3-7x^2+14x-8=0$$
where we test the following solutions of the fraction of the last coefficient divided by ...
- 2,973
5
votes
3
answers
188
views
Resolve: $4\sin(2x)+4\cos(x)-5=0$
The first thing that comes to mind is to substitute $\sin(2x)=2\sin(x)\cos(x)$ and so we have:
\begin{align*}
8\sin(x)\cos(x)+4\cos(x)-5=0
\end{align*}
But after that I can't see what other identity ...
- 59
1
vote
1
answer
302
views
Solving cubic equations with sine and cosine sums.
I was playing with math, and then I tried to rewrite some cubic equation with sine power reduction formula
$$y^3 + my^2 + ny + d = 0.$$
Let
$$y = \sin(x).$$
Then
$$y^2 = \frac{1 - \cos(2x)}{2},$$
$$y^...
1
vote
1
answer
47
views
How to solve the following cubic equation.
If the equation $3\beta sinx –1 = (\beta + sinx) (\beta^2 + sin^2x – \beta sinx)$, $\beta \in \mathbb{R}$ can be solved for x, then sum
of all possible integral values of $\beta$ .
First we can use ...
user930927
3
votes
2
answers
740
views
If $\sin^3(\theta)+\cos^3(\theta) = \frac{11}{16}$, find the exact value of $\sin(\theta) + \cos(\theta)$
The equation is
$$\sin^3(\theta)+\cos^3(\theta) = \frac{11}{16}$$ and it wants me to find the exact value of $\sin(\theta) + \cos(\theta)$.
I started at first trying to use Pythagorean identities, ...
1
vote
1
answer
139
views
How to solve for the exact answer of a cubic equation with very complicated constant term? (in order to solve for $\sin(1^\circ)$)
I am trying to figure out the exact value of $\sin(1^\circ)$ and $\cos(1^\circ)$ so that I am able to get the exact value of every integral sine and cosine. The list of formulas that we need are:
$$\...
- 477
3
votes
4
answers
135
views
Why $8^{\frac{1}{3}}$ is $1$, $\frac{2\pi}{3}$, and $\frac{4\pi}{3}$
The question is:
Use DeMoivre’s theorem to find $8^{\frac{1}{3}}$. Express your answer in complex form.
Select one:
a. 2
b. 2, 2 cis (2$\pi$/3), 2 cis (4$\pi$/3)
c. 2, 2 cis ($\pi$/3)
d. 2 cis ($\pi$/...
- 407
0
votes
1
answer
124
views
Trigonometric identity of $\cos \left (\frac {\theta}{3}\right)$
I was trying to solve a cubic equation using trigonometric representation of Cardano's Formula solutions.
My equation looks like this:
$$x^3-3mx+6m=0$$
I can only find $\cos(\theta)$ however to find ...
- 77
1
vote
2
answers
71
views
Analytically solving $\frac{1}{\sin2x} + \frac{1}{\sin3x} = \frac{1}{\sin x}$
Given
$$ \frac{1}{\sin(2x)} + \frac{1}{\sin(3x)} = \frac{1}{\sin x}$$
I tried solving the equation above using the double and triple angle formulas and arrived at this cubic expression in $\cos x$
$...
- 578
0
votes
0
answers
76
views
Solving $\beta^3-2\beta^2+(1-\rho)=0$ for $\beta$
I need to solve the following equation for $\beta$:
$$\beta^3-2\beta^2+(1-\rho)=0$$
where $\rho$ is just a constant.
I already tried different kind of methods from the wikipedia page for cubic ...
- 97
2
votes
1
answer
75
views
Verify triginometric result of cubic equation $x^3-x^2-p^2x+p^2=0$
Consider the following cubic function,
$$f(x):=(x+p)(x-p)(x-1)=x^3-x^2-p^2x+p^2$$
where $p\in(0,1)$ is a fixed parameter.
Then the sum of the absolute value of the three roots is
$$S_1:=1+2p$$
On the ...
- 740
1
vote
1
answer
63
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Estimating $f(x)=\frac{\sin(x)+\sec(x)+\tan(x)}{\cos(x)\csc(x)\cot(x)}$ where $x \in[-\frac{\pi}{3},\frac{\pi}{4}]$
Consider the function
$$f(x)=\frac{\sin(x)+\sec(x)+\tan(x)}{\cos(x)\csc(x)\cot(x)}$$
in the interval $x \in[-\frac{\pi}{3},\frac{\pi}{4}]$.
Find a combination of algebraic (not transcendental) ...
- 5,450