We can find the solutions of \sin x = 0.3 algebraically. First we find the solutions in the interval [0, 2 \pi). We get one such solution by taking \sin^{-1} to get x\approx______. We find all solutions by adding multiples of _____ to the solutions if [0, 2 \pi).

To determine
a)
To complete:
The statement "First we find the solutions in the interval ${\displaystyle [0,2\pi )}$. We get one such solution by taking ${\displaystyle 7{\sin }^{-1}}$ to
get ${\displaystyle x\approx }$ ______. The other solution in this interval is ${\displaystyle x\approx }$ ______.
Approach:
The range of the trigonometry function of ${\displaystyle \sin \theta }$ is lies between ${\displaystyle [-1,1]}$. No solution exists beyond this range. Sine has period ${\displaystyle 2\pi }$, we find solution in any interval of length ${\displaystyle 2\pi }$. Sine function is positive in first and second quadrant.
Calculation:
Consider the trigonometry equation.
${\displaystyle \sin \theta =0.3\dots ..\left(1\right)}$
Multiply ${\displaystyle {\sin }^{-1}}$ both side in equation (1).
${\displaystyle {\sin }^{-1}\left(\sin \theta \right)={\sin }^{-1}\left(0.3\right)}$
${\displaystyle {\theta }_1\approx 0.30}$
${\displaystyle {\theta }_2\approx 2.83}$
Therefore, the appropriate answers are ${\displaystyle {\theta }_1\approx 0.30}$ and ${\displaystyle {\theta }_2\approx 2.83}$.
Conclusion:
Thus, the appropriate answers are ${\displaystyle {\theta }_1\approx 0.30}$ and ${\displaystyle {\theta }_2\approx 2.83}$.
b) We find all solutions by adding multiples of _____ to the solutions if ${\displaystyle [0,2\pi )}$. The solutions are ${\displaystyle x\approx }$_______ and ${\displaystyle x\approx }$_______.
Approach:
Sine has period ${\displaystyle 2\pi }$, we find solution in any interval of length ${\displaystyle 2\pi }$. Sine function is positive in first and second quadrant.
The function repeats its value every ${\displaystyle 2\pi }$ units, so we get all solutions of the equation by adding integer multiples of ${\displaystyle 2\pi }$ to these solutions.
Calculation:
Consider the solutions.
${\displaystyle \theta \approx 0.30}$
${\displaystyle \theta \approx 2.83}$
The function repeats its value every ${\displaystyle 2\pi }$ units. So we get all solutions of the equation by adding integer multiples pf ${\displaystyle 2\pi }$ to these solutions.
${\displaystyle {\theta }_3\approx 0.30+2k\pi }$
${\displaystyle {\theta }_3\approx 2.83+2k\pi }$
Therefore, the appropriate answers are ${\displaystyle 2\pi ,{\theta }_3\approx 0.30+2k\pi }$ and ${\displaystyle {\theta }_4\approx 2.83+2k\pi }$
Conclusion:
Thus, the appropriate answers are ${\displaystyle 2\pi ,{\theta }_3\approx 0.30+2k\pi }$ and ${\displaystyle {\theta }_4\approx 2.83+2k\pi }$