I don't understand how to do these following equations:
Solve for x:
2(cos x)^2 = cos x for -(pi) < x < (pi)
cos 2x = 5sinx - 2 for all real x
I'm more interested in how to do them rather than the answers themselves.
Thank you very much, Adam
I don't understand how to do these following equations:
Solve for x:
2(cos x)^2 = cos x for -(pi) < x < (pi)
cos 2x = 5sinx - 2 for all real x
I'm more interested in how to do them rather than the answers themselves.
Thank you very much, Adam
Hi
$\displaystyle 2 \cos^2 x = \cos x$
$\displaystyle \cos x\: (2 \cos x -1) = 0$
$\displaystyle \cos x = 0$ or $\displaystyle \cos x = \frac{1}{2}$
$\displaystyle \cos 2x = 5 \sin x - 2$
$\displaystyle 1 - 2 \sin^2 x = 5 \sin x - 2$
$\displaystyle 2 \sin^2 x + 5 \sin x - 3 = 0$
$\displaystyle 2 X^2 + 5 X - 3 = 0$ with $\displaystyle X = \sin x$
$\displaystyle 2(\cos x)^2 = \cos x \ \ \ \ -\pi<x<\pi$
Let $\displaystyle u=\cos x$ and rearrange to get it into a quadratic equation form.
$\displaystyle 2u^2 - u=0$
This is in quadratic equation form as we have polynomial to power of
$\displaystyle 2$ thus we can factorise and solve for solutions.
$\displaystyle u(2u-1) = 0$
$\displaystyle u = 0$ or $\displaystyle 2u - 1= 0$
So your solutions will occur when $\displaystyle \cos x = 0$ and $\displaystyle \cos x = \frac{1}{2}$.
As $\displaystyle \cos x$ is a periodic function, thus there may be more than one solution of $\displaystyle x$ for between $\displaystyle -\pi<x<\pi$. One way to find the solution is to plot the graph between $\displaystyle -\pi<x<\pi$ for $\displaystyle \cos x$ and draw a line at $\displaystyle \frac{1}{2}$ and $\displaystyle 0$ and see where it intercepts the curve. Those would be you solutions.
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$\displaystyle \cos 2x = 5\sin x - 2 $
You need to know the double angle rule. For $\displaystyle \cos 2x$, there are three versions and you must pick the appropriate one. For your example, we would pick one involving $\displaystyle \sin x$. This would be $\displaystyle \cos 2x = 1 - 2\sin ^2 x$.
$\displaystyle \therefore 1 - 2\sin ^2 x = 5\sin x - 2$
Now you can let $\displaystyle u=\sin x$ and solve for solution of the quadratic equation and replace it with your substitution to find the specific values (as done in the first question).