How do I solve the equation in an interval from 0 to 2π ? 9 cos 2t = 6

Question

asked Dec 27, 2021 9.6k views

1 Answer

Zephinzer · Answer 1 · 2022-01-03T07:43:46+0000

$9\cos(2t)=6\implies\cos(2t)=\frac23$

Using the fact that cos is 2π-periodic, we have

$\cos(2t)=\frac23\implies2t=\cos^(-1)\left(\frac23\right)+2n\pi$

That is,
$\cos(\theta+2n\pi)=\cos\theta$ for any
$\theta$ and integer
.

$\implies t=\frac12\cos^(-1)\left(\frac23\right)+n\pi$

We get 2 solutions in the interval [0, 2π] for
n=0 and
n=1 ,

$t=\frac12\cos^(-1)\left(\frac23\right)\text{ and }t=\frac12\cos^(-1)\left(\frac23\right)+\pi$