Question

An equation is given. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.) 3 tan(3θ) − 1 = 0 (b) Find the solutions in the interval [0, 2π).

Answer 1

`3\tan3\theta-1=0`

`3\tan3\theta=1`

`\tan3\theta=\frac13`

Recall that the tangent function has a period of
`\pi` so that

`3\theta=\tan^(-1)\frac13+k\pi`

for any integer
`k`. Then

`\theta=\frac13\tan^(-1)\frac13+\frac{k\pi}3`

We get 6 solutions in the interval [0, 2π) for
`0\le k\le5`,

`\theta\approx0.107`

`\theta\approx1.154`

`\theta\approx2.202`

`\theta\approx3.249`

`\theta\approx4.296`

`\theta\approx5.343`