Exploring Half-Angle and Double-Angle Relations in Trigonometric Identities
Consider the trigonometric identity \(\sin(2\theta) = 2\sin\theta\cos\theta\) and its related form \(\cos\theta = 1 – 2\sin^2(\theta/2)\). (a) Derive an expression for \(\sin^2(\theta/2)\) in terms of \(\cos\theta\). (b) Using your derived formula, calculate \(\sin^2(\theta/2)\) when \(\cos\theta = 0.5\) and state the general solution for \(\theta\) that satisfies \(\cos\theta = 0.5\).