Trigonometrical Equations 3 Question 4

4. The number of values of $x$ in the interval $[0,5 \pi]$ satisfying the equation $3 \sin ^{2} x-7 \sin x+2=0$ is

(1998, 2M)

(a) 0

(b) 5

(c) 6

(d) 10

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Answer:

Correct Answer: 4. (c)

Solution:

  1. Given, $3 \sin ^{2} x-7 \sin x+2=0$

$\Rightarrow \quad 3 \sin ^{2} x-6 \sin x-\sin x+2=0$

$\Rightarrow \quad 3 \sin x(\sin x-2)-1(\sin x-2)=0$

$\Rightarrow \quad(3 \sin x-1)(\sin x-2)=0$

$\Rightarrow \quad \sin x=\frac{1}{3} \quad[\because \sin x=2$ is rejected $]$

$\Rightarrow \quad x=n \pi+(-1)^{n} \sin ^{-1} \frac{1}{3}, n \in I$

For $\quad 0 \leq n \leq 5, x \in[0,5 \pi]$

There are six values of $x \in[0,5 \pi]$ which satisfy the equation $3 \sin ^{2} x-7 \sin x+2=0$.



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