Chapter 1, Problem 58RE

(a)

To determine

To solve: Theequation $\sin x=-0.2$ for the interval $0\le x<2\pi$ .

Answer to Problem 58RE

The solution of the equation are $x\approx 3.3430$ and $x\approx 6.0818$ in the given interval.

Explanation of Solution

Given information:

The given function is $\sin x=-0.2$ in the interval $0\le x<2\pi$ .

Calculation:

Simplify the equation.

  $\begin{array}{c}\sin x=-0.2\\ x={\sin }^{-1}\left(-0.2\right)\end{array}$

Use the calculator to find the value of x . Enter the keystrokes given below:

  $\boxed{2\text{nd}}\boxed{\text{SIN}}\boxed{(-)}\boxed{0.2}\boxed{)}$

So, $x\approx -0.2014$ .

In the interval $\left[0,2\pi \right)$ , the solutions of the equation can be calculated as:

  $\begin{array}{c}x=\pi -{\sin }^{-1}\left(-0.2\right)\\ =\pi -\left(-0.2014\right)\\ \approx 3.343\end{array}$

And

  $\begin{array}{c}x={\sin }^{-1}\left(-0.2\right)+2\pi \\ =\left(-0.2014\right)+2\pi \\ \approx 6.0818\end{array}$

Therefore, the solution of the equation are $x\approx 3.3430$ and $x\approx 6.0818$ in the given interval.

(b)

To determine

To solve: The equation $\sin x=-0.2$ for the interval $-\infty <x<\infty$ .

Answer to Problem 58RE

Thesolution of the given equationare $x\approx 3.3430+2\pi k$ and $x\approx 6.0818+2\pi k$ for all $k\in \mathbb{Z}$ .

Explanation of Solution

Given information:The given equation is $\sin x=-0.2$ and the interval is $-\infty <x<\infty$ .

Calculation:

From part(a), the solution of equation are $x\approx 3.3430$ and $x\approx 6.0818$ .

It is known that the period of $\sin x$ is $2\pi$ .

So, the solutions of the equation $\sin x=-0.2$ in the interval $-\infty <x<\infty$ are:

  $x\approx 3.3430+2\pi k$ and $x\approx 6.0818+2\pi k$

For any integer $k$ .

Therefore,the value of the function is $x\approx 3.3430+2\pi k$ and $x\approx 6.0818+2\pi k$ for all $k\in \mathbb{Z}$ .