Chapter 7.4, Problem 4E

(a)

To determine

To fill: The blank in the statement “We can find the solutions of $\sin x=0.3$ algebraically.

First we find the solutions in the interval $\left[0,2\pi \right)$. We get one such solution by taking ${\sin }^{-1}$ to get $x\approx$________. The other solution in this interval is $x\approx$________”.

Answer to Problem 4E

The complete statement is “We can find the solutions of $\sin x=0.3$ algebraically. First we find the solutions in the interval $\left[0,2\pi \right)$. We get one such solution by taking ${\sin }^{-1}$ to get $\underset{¯}{x\approx 0.30}$. The other solution in this interval is $\underset{¯}{x\approx 2.84}$”.

Explanation of Solution

Consider $\sin x=0.3$.

Solve the above equation for x.

$\begin{array}{c}x={\sin }^{-1}\left(0.3\right)\\ =0.304\end{array}$

Find the other solution of the equation $\sin x=0.3$ as follows.

$\begin{array}{c}x=\pi -0.304\\ =2.84\end{array}$

Thus, the solutions of the equation $\sin x=0.3$ in the interval $\left[0,2\pi \right)$ are $x=0.30,\text{ and }2.84$.

Hence, the complete statement is “We can find the solutions of $\sin x=0.3$ algebraically. First we find the solutions in the interval $\left[0,2\pi \right)$. We get one such solution by taking ${\sin }^{-1}$ to get $\underset{¯}{x\approx 0.30}$. The other solution in this interval is $\underset{¯}{x\approx 2.84}$”.

(b)

To determine

To fill: The blank in the statement “We can find the solutions of $\sin x=0.3$ algebraically.

We find all solutions by adding multiples of _________ to the solutions in $\left[0,2\pi \right)$. The solutions are $x\approx$________ and $x\approx$________”.

Answer to Problem 4E

The complete statement is “We can find the solutions of $\sin x=0.3$ algebraically. We can find all solutions by adding multiples of $\underset{¯}{2\pi }$ to the solutions in $\left[0,2\pi \right)$. The solutions are $\underset{¯}{x\approx 0.30+2k\pi }$ and $\underset{¯}{x\approx 2.84+2k\pi }$”.

Explanation of Solution

Consider the equation $\sin x=0.3$.

From part (a), it is obtained that its solutions in the interval $\left[0,2\pi \right)$ are $x=0.30,\text{ and }2.84$.

As the sine function is a periodic function of period $2\pi$, add integer multiples of $2\pi$ to the solutions in the interval $\left[0,2\pi \right)$.

Hence, the solutions are $x\approx 0.30+2k\pi$ and $x\approx 2.84+2k\pi$.

Hence, the complete statement is “We can find the solutions of $\sin x=0.3$ algebraically. We can find all solutions by adding multiples of $\underset{¯}{2\pi }$ to the solutions in $\left[0,2\pi \right)$. The solutions are $\underset{¯}{x\approx 0.30+2k\pi }$ and $\underset{¯}{x\approx 2.84+2k\pi }$”.