Chapter 2.1, Problem 79E

(a)

To determine

To find: The value of $f\left(\frac{\pi }{6}\right)$ .

Answer to Problem 79E

The value of $f\left(\frac{\pi }{6}\right)$ is $\frac{1}{2}$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=\sin x$ .

Calculation:

Substitute $\frac{\pi }{6}$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=\sin x\\ f\left(\frac{\pi }{6}\right)=\sin \frac{\pi }{6}\\ f\left(\frac{\pi }{6}\right)=\frac{1}{2}\end{array}$

Therefore, the value $f\left(\frac{\pi }{6}\right)$ is $\frac{1}{2}$ .

(b)

To determine

To find: The interval $\left(a,b\right)$ about $x=\frac{\pi }{6}$ so that $0.3<f\left(x\right)<0.7$ with the help of graph provided $a<x<b$ .

Answer to Problem 79E

The value of $a$ and $b$ are $0.3047$ and $0.7754$ respectively.

Explanation of Solution

Given information: The function is $f\left(x\right)=\sin x$ .

Calculation:

The point $a$ is the intersection of the curve $f\left(x\right)=\sin x$ and $y=0.3$ , the point $b$ is the intersection of the curve $f\left(x\right)=\sin x$ and $y=0.7$ .

Graph the curves.

To graph a function $f\left(x\right)=\sin x$ , $y=0.3$ and $y=0.7$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $f\left(x\right)=\sin x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{SIN}}\boxed{\text{X}}\boxed{)}$ . Now press Enter, enter right hand side of the equation $y=0.3$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{0.3}$ . Now press Enter, enter right hand side of the equation $y=0.7$ after the symbol ${\text{Y}}_3$ . Enter the keystrokes $\boxed{0.7}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\sin \left(\text{X}\right)\\ {\text{Y}}_2=0.3\\ {\text{Y}}_3=0.7\end{array}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

The curves $f\left(x\right)=\sin x$ and $y=0.3$ intersects at the point $x=0.3047$ and the curves $f\left(x\right)=\sin x$ and $y=0.7$ intersects at the point $x=0.7754$ .

Therefore, the value of $a$ and $b$ is $0.3047$ and $0.7754$ respectively.

(c)

To determine

To find: The interval $\left(a,b\right)$ so that $0.49<f\left(x\right)<0.51$ with the help of graph provided $a<x<b$ .

Answer to Problem 79E

The value of $a$ and $b$ are $0.5121$ and $0.5352$ respectively.

Explanation of Solution

Given information: The function is $f\left(x\right)=\sin x$ .

Calculation:

The point $a$ is the intersection of the curve $f\left(x\right)=\sin x$ and $y=0.49$ , the point $b$ is the intersection of the curve $f\left(x\right)=\sin x$ and $y=0.51$ .

Graph the curves.

To graph a function $f\left(x\right)=\sin x$ , $y=0.49$ and $y=0.51$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $f\left(x\right)=\sin x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{SIN}}\boxed{\text{X}}\boxed{)}$ . Now press Enter, enter right hand side of the equation $y=0.49$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{0.49}$ . Now press Enter, enter right hand side of the equation $y=0.51$ after the symbol ${\text{Y}}_3$ . Enter the keystrokes $\boxed{0.51}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\sin \left(\text{X}\right)\\ {\text{Y}}_2=0.49\\ {\text{Y}}_3=0.51\end{array}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

The curves $f\left(x\right)=\sin x$ and $y=0.49$ intersects at the point $x=0.5121$ and the curves $f\left(x\right)=\sin x$ and $y=0.51$ intersects at the point $x=0.5352$ .

Therefore, the value of $a$ and $b$ is $0.5121$ and $0.5352$ respectively.