Chapter 4.2, Problem 71E

(a)

To determine

To calculate: The value of $\sin 5$ using the given figure and a straightedge and then to check the approximation using a graphing utility.

Answer to Problem 71E

The value of $\sin 5$ is approximately equal to $-0.95$ .

Explanation of Solution

Given information:A figure is provided.

Calculation: Assume that any point on the provided circle of unit length be denoted by $\left(x,y\right)$ , where $x=\cos \theta \text{ and }y=\sin \theta$ .

• Here, $\theta$ is the arc length or the angle measured in radians and 3’o clock from the standard position.

Now, to calculate $\sin 5$ , observe that $\theta =5$ .

The point that can be pointed on the circle is approximately equal to $\left(0.3,-0.95\right)$ .

Therefore,

  $\begin{array}{c}\sin 5=y\\ =-0.95\end{array}$

The graph using a graphing utility as follows:

  

The value can be approximated (in radians) using a calculator as,

  $\begin{array}{c}\sin 5=-0.95892427466\\ \approx -0.95\end{array}$

Hence, $\sin 5\approx -0.95$ .

(b)

To determine

To calculate: The value of $\cos 2$ using the given figure and a straightedge and then to check the approximation using a graphing utility.

Answer to Problem 71E

The value of $\cos 2$ is approximately equal to $-0.4$ .

Explanation of Solution

Given information: A figure is provided.

Calculation: Assume that any point on the provided circle of unit length be denoted by $\left(x,y\right)$ , where $x=\cos \theta \text{ and }y=\sin \theta$ .

• Here, $\theta$ is the arc length or the angle measured in radians and 3’o clock from the standard position.

Now, to calculate $\cos 2$ , observe that $\theta =2$ .

The point that can be pointed on the circle is approximately equal to $(-0.4,0.9)$ .

Therefore,

  $\begin{array}{c}\cos 2=x\\ =-0.4\end{array}$

The graph using a graphing utility as follows:

  

The value can be approximated (in radians) using a calculator as,

  $\begin{array}{c}\cos 2=-0.41614683654\\ \approx -0.4\end{array}$

Hence, $\cos 2\approx -0.4$ .