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 \(\cos\theta = 0.5\).
Solving a Half-Angle Equation for Exact Angles
Determine all angles \(\theta\) in the interval \([0, 2\pi]\) that satisfy the equation \(\sin\left(\frac{\theta}{2}\right) = \frac{1}{2}\). Provide your answers as exact values.
Derivation of sin(π/8) Using the Half-Angle Formula
Using the known value of \(\sin(\pi/4)\) and the half-angle identity, derive the exact value of \(\sin(\pi/8)\). Show all necessary steps in your derivation.