Chapter 7.5, Problem 60E

To determine

To find: possible number of solutions for the given equation.

Answer to Problem 60E

  $2b$

Explanation of Solution

Given information:

  $\begin{array}{c}a\sin \left(bx+c\right)+d=d+\frac{a}{2}\\ 0°\le x<360°\\ a\ne 0\\ b=\text{ positive integer}\end{array}$

Calculation:

Simplify the equation as shown below.

  $\begin{array}{c}a\sin \left(bx+c\right)+d=d+\frac{a}{2}\\ a\sin \left(bx+c\right)=\frac{a}{2}\end{array}$

If $x\to 0\text{ to }2\pi$ then $bx+c\to c\text{ to }2b\pi +c$ , covering the range of $2b\pi$ . Since b is a positive integer, the range can go from $2\pi +c,4\pi +c,6\pi +c$ and so on. Its worth nothing that they are increments of full periods.

Now, any sine will return value from $-1\text{ to }1$ . Since it is multiplied by $a$ , the function will return values from $-a\text{ to }a$ . The function will have exactly two times the value of $\frac{a}{2}$ for each period. The one full period is covered by $b$ times. Thus, the possible number of solutions is $2b$ .