Chapter 4.3, Problem 86E

a.

To determine

Complete the table.

Answer to Problem 86E

  $\boxed{\sin 0.1=0.0998}$ , $\boxed{\sin 0.2=0.1987}$ , $\boxed{\sin 0.3=0.2955}$ , $\boxed{\sin 0.4=0.3894}$ and $\boxed{\sin 0.5=0.4794}$ .

Explanation of Solution

Given information:

Complete the table.

  $\begin{array}{cccccc}\theta & 0.1& 0.2& 0.3& 0.4& 0.5\\ \sin \theta & & & & & \end{array}$

Calculation:

We shall find the corresponding value of $\sin \theta$ and hence complete the following table.

Let us use a calculator to find these values.

For $\theta =0.1$ ,

Since the angle is givem in radians, this, we will first set the mode of the calculator to radian mode.

The calculator keystroke is as follows,

  $\boxed{SIN}$$\boxed{(}0.1\boxed{)}$$\boxed{ENTER}$ .

Hence, using a calculator, the value of

  $\boxed{\sin 0.1=0.0998}$

For $\theta =0.2$ ,

Since the angle is given in radians, thus, we will first set the mode of the calculator to radian mode.

The calculator keystroke sequence is as follows,

  $\boxed{SIN}$$\boxed{(}0.2\boxed{)}$$\boxed{ENTER}$

Hence, using a calculator, the value of $\boxed{\sin 0.2=0.1987}$ .

For $\theta =0.3$ ,

Since the angle is given in radians, thus, we will first set the mode of the calculator to radian mode.

The calculator keystroke sequence is as follows,

  $\boxed{SIN}$$\boxed{(}0.3\boxed{)}$$\boxed{ENTER}$

Hence, using a calculator, the value of $\boxed{\sin 0.3=0.2955}$ .

For $\theta =0.4$ ,

Since the angle is given in radians, thus, we will first set the mode of the calculator to radian mode.

The calculator keystroke sequence is as follows,

  $\boxed{SIN}$$\boxed{(}0.4\boxed{)}$$\boxed{ENTER}$

Hence, using a calculator, the value of $\boxed{\sin 0.4=0.3894}$ .

For $\theta =0.5$ ,

Since the angle is given in radians, thus, we will first set the mode of the calculator to radian mode.

The calculator keystroke sequence is as follows,

  $\boxed{SIN}$$\boxed{(}0.5\boxed{)}$$\boxed{ENTER}$

Hence, using a calculator, the value of $\boxed{\sin 0.5=0.4794}$ .

b.

To determine

Is $\theta$ or $\sin \theta$ greater for in the interval $(0,0.5]$ ?

Answer to Problem 86E

  $\theta$ is slightly greater than the corresponding $\sin \theta$ .

Explanation of Solution

Given information:

Is $\theta$ or $\sin \theta$ greater for in the interval $(0,0.5]$ ?

Calculation:

For $\theta =0.1$ , $\sin \theta =0.0998$ .

For $\theta =0.2$ , $\sin \theta =0.1987$ .

For $\theta =0.3$ , $\sin \theta =0.2955$ and so on.

Hence for the interval $(0,5]$ , $\theta$ is slightly greater than the corresponding $\sin \theta$ .

c.

To determine

How do $\theta$ and $\sin \theta$ compare?

Answer to Problem 86E

  $\boxed{\frac{\theta }{\sin \theta }\to 1}$

Explanation of Solution

Given information:

As $\theta$ approaches $0$ , how do $\theta$ and $\sin \theta$ compare? Explain.

Calculation:

Now, from above we see that as $\theta \to 0\sin \theta \to 0$ and hence $\frac{\theta }{\sin \theta }\to 1$ .