Euler’s Formula and Compound Angles
Using Euler’s formula, derive the compound angle identities for sine and cosine. Start from the identity \(e^{i(A+B)} = e^{iA}e^{iB}\) and explain how comparing the real and imaginary parts leads to \(\cos(A+B)=\cos A\cos B-\sin A\sin B\) and \(\sin(A+B)=\sin A\cos B+\cos A\sin B\).
Solving Angle Equations Using the Double-Angle Tangent Identity
Solve for \(\theta\) in degrees, where \(0^\circ \leq \theta < 180^\circ\), given that \(\tan(2\theta) = \sqrt{3}\).