Chapter 14.2, Problem 71PPSE

Solution

To find: The condition on variables a , b of given trigonometric equation has exactly two solutions in given interval.

The option ( D ) is correct.

Given data: The given trigonometric equation is $a\sin \theta =b,a>0$ and given interval is $0\le \theta <2\pi$ , select one answer from the following:

  $(A)a=b(B)b>a(C)a=-b(D)a>b>-a$

Method/Formula used:

The values of $\sin \theta$ lie between -1 and 1.

Calculation:

The given trigonometric equation is

  $a\sin \theta =b,a>0$ … (1)

Solve equation for $\sin \theta$ ,

  $\sin \theta =\frac{b}{a}$

Since value of $\sin \theta$ lies between -1 and 1, therefore,

  $-1<\sin \theta <1$ … (3)

Substitute $\frac{b}{a}$ for $\sin \theta$ in condition (3),

  $\begin{array}{c}-1<\frac{b}{a}<1\underset{\downarrow }{\underbrace{\text{Multiply}\text{by}\text{a}}}\\ -1\cdot a<\frac{b}{a}\cdot a<1\cdot a\\ -a<b<a\\ \text{OR}a>b>-a\end{array}$

Therefore, the option ( D ) is correct.