Chapter 7.6, Problem 10E

Interpretation Introduction

Interpretation:

To derive the identity ${\text{sinθcos}}^{\text{2}}\text{θ = }\frac{\text{1}}{\text{4}}\left[\text{sinθ + sin3θ}\right]$ using complex representation $\text{cosθ = }\frac{{\text{e}}^{\text{iθ}}{\text{+e}}^{\text{-iθ}}}{\text{2}}$ and $\text{sinθ = }\frac{{\text{e}}^{\text{iθ}}{\text{- e}}^{\text{-iθ}}}{\text{2i}}$.

Concept Introduction:

The complex representation for cosine is $\text{cosθ = }\frac{{\text{e}}^{\text{iθ}}{\text{+e}}^{\text{-iθ}}}{\text{2}}$ and for sine is $\text{sinθ = }\frac{{\text{e}}^{\text{iθ}}{\text{- e}}^{\text{-iθ}}}{\text{2i}}$.

Multiplying ${\text{cos}}^{\text{2}}\text{θ}$ by $\text{sinθ}$ will give required identity.