Question

find all solutions of the equation in the interval $0,2\pi)$.
$\sin\theta = 0.5767$
if there is more than one solution, separate them with commas.
do not round any intermediate computations. give your answer(s) in radians, and round your answer(s) to the nearest hundredth.
$\theta =$

Explanation:

Step1: Find the principal solution

We know that if \(\sin\theta = x\), then \(\theta=\sin^{- 1}(x)+2k\pi\) or \(\theta=\pi-\sin^{-1}(x)+2k\pi\), \(k\in\mathbb{Z}\).
For \(x = 0.5767\), the principal solution \(\theta_1=\sin^{-1}(0.5767)\). Using a calculator, \(\sin^{-1}(0.5767)\approx0.61\) radians.

Step2: Find the second - quadrant solution

The second - quadrant solution is \(\theta_2=\pi-\sin^{-1}(0.5767)\).
Since \(\pi\approx3.14\), \(\theta_2 = 3.14 - 0.61=2.53\) radians.

Answer:

\(0.61,2.53\)