# 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).

We can find the solutions of $\mathrm{sin}x=0.3$ algebraically.
a) First we find the solutions in the interval $\left[0,2\pi \right)$. We get one such solution by taking ${\mathrm{sin}}^{-1}$ to get $x\approx$______. The other solution in this interval is $x\approx$______.
b) We find all solutions by adding multiples of _____ to the solutions if $\left[0,2\pi \right)$. The solutions are $x\approx$_______ and $x\approx$_______.
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To determine
a)
To complete:
The statement "First we find the solutions in the interval $\left[0,2\pi \right)$. We get one such solution by taking $7{\mathrm{sin}}^{-1}$ to
get $x\approx$ ______. The other solution in this interval is $x\approx$ ______.
Approach:
The range of the trigonometry function of $\mathrm{sin}\theta$ is lies between $\left[-1,1\right]$. No solution exists beyond this range. Sine has period $2\pi$, we find solution in any interval of length $2\pi$. Sine function is positive in first and second quadrant.
Calculation:
Consider the trigonometry equation.
$\mathrm{sin}\theta =0.3\dots ..\left(1\right)$
Multiply ${\mathrm{sin}}^{-1}$ both side in equation (1).
${\mathrm{sin}}^{-1}\left(\mathrm{sin}\theta \right)={\mathrm{sin}}^{-1}\left(0.3\right)$
${\theta }_{1}\approx 0.30$
${\theta }_{2}\approx 2.83$
Therefore, the appropriate answers are ${\theta }_{1}\approx 0.30$ and ${\theta }_{2}\approx 2.83$.
Conclusion:
Thus, the appropriate answers are ${\theta }_{1}\approx 0.30$ and ${\theta }_{2}\approx 2.83$.
b) We find all solutions by adding multiples of _____ to the solutions if $\left[0,2\pi \right)$. The solutions are $x\approx$_______ and $x\approx$_______.
Approach:
Sine has period $2\pi$, we find solution in any interval of length $2\pi$. Sine function is positive in first and second quadrant.
The function repeats its value every $2\pi$ units, so we get all solutions of the equation by adding integer multiples of $2\pi$ to these solutions.
Calculation:
Consider the solutions.
$\theta \approx 0.30$
$\theta \approx 2.83$
The function repeats its value every $2\pi$ units. So we get all solutions of the equation by adding integer multiples pf $2\pi$ to these solutions.
${\theta }_{3}\approx 0.30+2k\pi$
${\theta }_{3}\approx 2.83+2k\pi$
Therefore, the appropriate answers are $2\pi ,{\theta }_{3}\approx 0.30+2k\pi$ and ${\theta }_{4}\approx 2.83+2k\pi$
Conclusion:
Thus, the appropriate answers are $2\pi ,{\theta }_{3}\approx 0.30+2k\pi$ and ${\theta }_{4}\approx 2.83+2k\pi$