2. Show that if \( \cos (\theta+30)=1 / 2 \sin \theta \) then \( \tan \theta=\frac{\sqrt{3}}{2} \)

Solve the equation \[ \cos(\theta+30^\circ)=\frac{1}{2}\sin\theta \] step by step. 1. Write \(\cos(\theta+30^\circ)\) using the cosine addition formula: \[ \cos(\theta+30^\circ)=\cos\theta\cos30^\circ-\sin\theta\sin30^\circ. \] 2. Substitute the known values \(\cos30^\circ=\frac{\sqrt{3}}{2}\) and \(\sin30^\circ=\frac{1}{2}\): \[ \cos(\theta+30^\circ)=\cos\theta\left(\frac{\sqrt{3}}{2}\right)-\sin\theta\left(\frac{1}{2}\right). \] 3. Set the expression equal to \(\frac{1}{2}\sin\theta\) as given: \[ \cos\theta\left(\frac{\sqrt{3}}{2}\right)-\sin\theta\left(\frac{1}{2}\right)=\frac{1}{2}\sin\theta. \] 4. Multiply the entire equation by 2 to eliminate the denominators: \[ \sqrt{3}\cos\theta-\sin\theta=\sin\theta. \] 5. Add \(\sin\theta\) to both sides: \[ \sqrt{3}\cos\theta=2\sin\theta. \] 6. Divide both sides by \(\cos\theta\) (assuming \(\cos\theta\neq0\)): \[ \sqrt{3} = 2\tan\theta. \] 7. Solve for \(\tan\theta\): \[ \tan\theta=\frac{\sqrt{3}}{2}. \] Thus, we have shown that if \[ \cos(\theta+30^\circ)=\frac{1}{2}\sin\theta, \] then \[ \tan\theta=\frac{\sqrt{3}}{2}. \]