7. Find the general solution of each of the following equations: \( \begin{array}{ll}\text { (a) } \cos 2 \theta=0,4 & \text { (b) } 5 \sin \theta+14 \cos \theta=0\end{array} \)
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To solve the equation \( \cos 2\theta = 0.4 \), we can use the double angle identity \( \cos 2\theta = 2\cos^2 \theta - 1 \). Rearranging, we have \( 2\cos^2 \theta = 1.4 \) which simplifies to \( \cos^2 \theta = 0.7 \). Thus, \( \cos \theta = \pm \sqrt{0.7} \). The general solutions for \( \theta \) are given by \( \theta = \cos^{-1}(\sqrt{0.7}) + 2k\pi \) and \( \theta = \cos^{-1}(-\sqrt{0.7}) + 2k\pi \), where \( k \) is any integer. For the second equation \( 5\sin \theta + 14\cos \theta = 0 \), we can rearrange this to find \( \tan \theta \) by isolating \( \sin \theta \): \( \sin \theta = -\frac{14}{5} \cos \theta \). Dividing both sides by \( \cos \theta \) gives \( \tan \theta = -\frac{14}{5} \). The general solutions are \( \theta = \tan^{-1}\left(-\frac{14}{5}\right) + n\pi \), where \( n \) is any integer.
