Question

The general solution of the equation \[7 \cos^2 x + 3 \sin^2 x = 4\] is

Options

• \[x = 2 n\pi \pm \frac{\pi}{6}, n \in Z\]

   

• \[x = 2 n\pi \pm \frac{2\pi}{3}, n \in Z\]

   

• \[x = n\pi \pm \frac{\pi}{3}, n \in Z\]

• none of these

Answer 1

\[x = n\pi \pm \frac{\pi}{3}, n \in Z\]
Given:
\[7 \cos^2 x + 3 \sin^2 x = 4\]
\[ \Rightarrow 7 \cos^2 x + 3 (1 - \cos^2 x) = 4\]
\[ \Rightarrow 7 \cos^2 x + 3 - 3 \cos^2 x = 4\]
\[ \Rightarrow 4 \cos^2 x + 3 = 4\]
\[ \Rightarrow 4 (1 - \cos^2 x) = 3\]
\[ \Rightarrow 4 \sin^2 x = 3\]
\[ \Rightarrow \sin^2 x = \frac{3}{4}\]
\[ \Rightarrow \sin x = \frac{\sqrt{3}}{2}\]
\[ \Rightarrow \sin x = \sin \frac{\pi}{3}\]
\[ \Rightarrow x = n\pi \pm \frac{\pi}{3}, n \in Z\]