Question

Solve the equation in the interval 0 to 2pi.

Answer 1

Answer:

You don't need to single out $\cos \theta$ to solve this question.  

To solve:

$\begin{aligned}3 \cos 4 \theta & = -2\\\\\cos 4 \theta & = -\dfrac{2}{3}\\\\4 \theta & = \cos^{-1}\left( -\dfrac{2}{3\right)}\\\\\implies 4 \theta & =2.30053... \pm 2 \pi n, -2.30053...\pm 2 \pi n\\\\\theta & =\dfrac{2.30053...}{4} \pm \dfrac{\pi n}{2}, -\dfrac{2.30053...}{4}\pm \dfrac{\pi n}{2}\end{aligned}$

So for the given interval $0\leq \theta \leq 2 \pi$

$\implies \theta =0.575, 2.146, 3.717, 5.288, 0.996, 2.566, 4.137, 5.708\:\:(\sf 3 \: d.p.)$

(As confirmed by the attached graph)

Answer 2

$\\ \rm\Rrightarrow 3cos4\theta=-2$

$\\ \rm\Rrightarrow cos4\theta=\dfrac{-2}{3}$

$\\ \rm\Rrightarrow 4\theta=cos^{-1}\left(\dfrac{-2}{3}\right)$

$\\ \rm\Rrightarrow \theta=\pm\dfrac{cos^{-1}\left(\dfrac{-2}{3}\right)}{4}$

Using scientific calculator

$\\ \rm\Rrightarrow \theta=\pm 32.95257+90°n$

So as interval is [0,2π] 4 values are there

• For 90°

$\\ \rm\Rrightarrow \theta=32.95257+90°= 122.95277°$

$\\ \rm\Rrightarrow \theta=-32.95257+90=57.04743°$

• For 2π-90=270°

$\\ \rm\Rrightarrow \theta=32.95257+270=302.95257°$

$\\ \rm\Rrightarrow \theta=32.95257+270=237.04743°$