The number of real solutions in the interval −3π2,7π2 of the equation 4sin2⁡x−8sin⁡x+3=0 is

4sin2⁡x−8sin⁡x+3=0⇒4sin2⁡x−6sin⁡x−2sin⁡x+3=0⇒(2sin⁡x−3)(2sin⁡x−1)=0⇒sin⁡x=12⇒sin⁡x=sin⁡π6⇒x=nπ+(−1)nπ6x∈−3π2,7π2⇒x=−7π6,π6,5π6,13π6,17π6. ∴ The number of real solutions =5

The number of real solutions in the interval −3π2,7π2 of the equation 4sin2⁡x−8sin⁡x+3=0 is

4sin2⁡x−8sin⁡x+3=0⇒4sin2⁡x−6sin⁡x−2sin⁡x+3=0⇒(2sin⁡x−3)(2sin⁡x−1)=0⇒sin⁡x=12⇒sin⁡x=sin⁡π6⇒x=nπ+(−1)nπ6x∈−3π2,7π2⇒x=−7π6,π6,5π6,13π6,17π6. ∴ The number of real solutions =5

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The number of real solutions in the interval $[- \frac{3 π}{2}, \frac{7 π}{2}]$ of the equation $4 \sin^{2} x - 8 \sin x + 3 = 0$ is

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Detailed Solution

$\begin{array}{l} 4 \sin^{2} x - 8 \sin x + 3 = 0 \Rightarrow 4 \sin^{2} x - 6 \sin x - 2 \sin x + 3 = 0 \\ \Rightarrow (2 \sin x - 3) (2 \sin x - 1) = 0 \Rightarrow \sin x = \frac{1}{2} \Rightarrow \sin x = \sin \frac{π}{6} \Rightarrow x = n π + (- 1)^{n} \frac{π}{6} \\ x \in [\frac{- 3 π}{2}, \frac{7 π}{2}] \Rightarrow x = \frac{- 7 π}{6}, \frac{π}{6}, \frac{5 π}{6}, \frac{13 π}{6}, \frac{17 π}{6} . ∴ The number of real solutions = 5 \end{array}$