Question

Solve the equation in the interval 0 to 2pi.

3 Cos 4θ = -2

I keep trying to single out Cos θ, but once I get to 4θ I'm stuck, the closest to an answer I got was -1/6 but I dont know what that would be on the interval 0 to 2pi​

Answer 1

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

`\\ \rm\Rrightarrow cos4\theta=(-2)/(3)`

`\\ \rm\Rrightarrow 4\theta=cos^(-1)\left((-2)/(3)\right)`

`\\ \rm\Rrightarrow \theta=\pm(cos^(-1)\left((-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°`

Answer 2

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 & = -(2)/(3)\\\\4 \theta & = \cos^(-1)\left( -(2)/(3\right))\\\\\implies 4 \theta & =2.30053... \pm 2 \pi n, -2.30053...\pm 2 \pi n\\\\\theta & =(2.30053...)/(4) \pm (\pi n)/(2), -(2.30053...)/(4)\pm (\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)