Question

Verify
sin(180degrees-θ)=sin θ

Answer 1

Explanation:

sin(x + y) = sin(x)×cos(y) + cos(x)×sin(y)

sin(x - y) = sin(x)×cos(y) - cos(x)×sin(y)

x = 180°

y = theta

sin(180°) = 0

cos(180°) = -1

sin(180° - theta) = sin(180°)×cos(theta) -

cos(180°)×sin(theta) =

= 0×cos(theta) - -1×sin(theta) =

= 0 + sin(theta) = sin(theta)

Answer 2

`\textit{Sum and Difference Identities} \\\\ \sin(\alpha + \beta)=\sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) \\\\ \sin(\alpha - \beta)=\sin(\alpha)\cos(\beta)- \cos(\alpha)\sin(\beta) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \sin(180^o-\theta )~~ = ~~\sin(\theta ) \\\\[-0.35em] ~\dotfill\\\\ \sin(180^o-\theta )\implies \sin(180^o)\cos(\theta )-\cos(180^o)\sin(\theta ) \\\\\\ \text{\LARGE 0}\cos(\theta )-\text{\LARGE (-1)}\sin(\theta )\implies \sin(\theta ) ~~ \checkmark`