# The number of value of $x$ in the interval $\left [ 0,3\pi \right ]$  satisfying the equation $2\sin ^{2}x+5\sin x-3= 0$is  Option 1) $4$ Option 2) $6$ Option 3) $1$ Option 4) $2$

P Plabita

As we learnt in

Trigonometric Equations -

The equations involving trigonometric function of unknown angles are known as trigonometric equations.

- wherein

e.g. $\cos ^{2}\Theta - 4\cos \Theta = 1$

$2\sin ^2 x + 5\sin x-3=0$   $\Rightarrow \sin x= \frac{-5 \pm \sqrt{25+24}}{4}$

$\Rightarrow \sin x= \frac{-5 \pm 7}{4}$   $\Rightarrow \sin x = \frac{1}{2}$                      $\left[sin\ x \neq -3]$

Now, in rage $\left [ 0,3\pi \right ]$

$\sin x=\frac{1}{2}$   is satisfied by

$x = \frac{\pi }{6}, \frac{5\pi }{6}, \frac{13\pi }{6},\frac{17\pi }{6}$

4 solution

Option 1)

$4$

Correct

Option 2)

$6$

Incorrect

Option 3)

$1$

Incorrect

Option 4)

$2$

Incorrect

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