Question

Se an Addition or Subtraction Formula to simplify the equation. sin θ cos 3θ + cos θ sin 3θ = 0

Answer 1

`\bf \textit{Sum and Difference Identities} \\ \quad \\ sin({{ \alpha}} + {{ \beta}})=sin({{ \alpha}})cos({{ \beta}}) + cos({{ \alpha}})sin({{ \beta}}) \\ \quad \\ sin({{ \alpha}} - {{ \beta}})=sin({{ \alpha}})cos({{ \beta}})- cos({{ \alpha}})sin({{ \beta}})\\\\ -------------------------------\\\\ sin(\theta )cos(3\theta )+cos(\theta )sin(3\theta )=0\implies sin(\theta ~+~3\theta )=0`


`\bf sin(4\theta )=0\implies 4\theta =sin^(-1)(0)\implies 4\theta = \begin{cases} 0\\ \pi \\ 2\pi \end{cases}\\\\ -------------------------------\\\\ 4\theta =0\implies \measuredangle \theta =0\\\\ -------------------------------\\\\ 4\theta =\pi \implies \measuredangle \theta =\cfrac{\pi }{4}\\\\ -------------------------------\\\\ 4\theta =2\pi \implies \measuredangle \theta =\cfrac{2\pi }{4}\implies \measuredangle \theta =\cfrac{\pi }{2}`