Chapter 10.5, Problem 22E

Solution

To find: A sine function with given minimum and maximum value.

Required function is $y\left(x\right)=5\sin 2x+3$ .

Given data:

The sine function has minimum at $\left(\frac{\pi }{2},3\right)$ and maximum at $\left(\frac{\pi }{4},8\right)$

Method/Formula used:

A sine function can be written as:

  $y\left(x\right)=A\sin Bx+C$ .

Calculation:

Suppose the required sine function is

  $y\left(x\right)=A\sin Bx+C$ … (1)

The minimum and maximum values of the sine function are at $\left(\frac{\pi }{2},3\right)$ and $\left(\frac{\pi }{4},8\right)$ respectively.

Therefore, the graph of the function passes through points $\left(\frac{\pi }{2},3\right)$ and $\left(\frac{\pi }{4},8\right)$ .

Substitute, $\frac{\pi }{2}$ for x and 3 for y in result (1),

  $3=A\sin \left(B\frac{\pi }{2}\right)+C$ … (2)

Substitute, $\frac{\pi }{4}$ for x and 8 for y in result (1),

  $8=A\sin \left(B\frac{\pi }{4}\right)+C$ … (3)

Minimum value of a sine function occurs at $x=0$ or $x=\pi$ . From this condition $x=\pi$ , therefore,

  $\begin{array}{c}B\frac{\pi }{2}=\pi \\ B=2\end{array}$

Maximum value of a sine function occurs at $x=\frac{\pi }{2}$ . Therefore,

  $\begin{array}{c}B\frac{\pi }{4}=\frac{\pi }{2}\\ B=2\end{array}$

Thus, suitable value of B is 2.

Substitute 2 for B in result (2),

  $\begin{array}{c}3=A\sin \left(\pi \right)+C\\ 3=A\cdot 0+C\\ 3=C\end{array}$

Substitute 2 for B in result (3),

  $\begin{array}{c}8=A\sin \left(2\frac{\pi }{4}\right)+C\left[\because \sin \left(\frac{\pi }{2}\right)=1\right]\\ =A+C\end{array}$

Substitute 3 for C

  $\begin{array}{c}A+C=8\\ A+3=8\\ A=8-3\\ A=5\end{array}$

Substitute 5 for A , 2 for B and 3 for C in result (1),

  $y\left(x\right)=5\sin 2x+3$