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

$3\tan3\theta-1=0$

$3\tan3\theta=1$

$\tan3\theta=\dfrac13$

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

$3\theta=\tan^{-1}\dfrac13+k\pi$

for any integer $k$. Then

$\theta=\dfrac13\tan^{-1}\dfrac13+\dfrac{k\pi}3$

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

$\theta\approx0.107$

$\theta\approx1.154$

$\theta\approx2.202$

$\theta\approx3.249$

$\theta\approx4.296$

$\theta\approx5.343$