Question

Rewrite the following as a product of trigonometric functions:sin 3° + sin 37°

Answer 1

Recall the following sum-to-product formulas of trigonometric function

`\sin \alpha+\sin \beta=2\sin \mleft((\alpha+\beta)/(2)\mright)\cos \mleft((\alpha−\beta)/(2)\mright)`

Given the sum

`\begin{gathered} \sin 3\degree+\sin 37\degree \\ \\ \alpha=3\degree \\ \beta=37\degree \end{gathered}`

Then the product is

`\begin{gathered} \sin 3\degree+\sin 37\degree=2\sin \mleft((3\degree+37\degree)/(2)\mright)\cos \mleft((3\degree-37\degree)/(2)\mright) \\ \\ \text{Simplifying we get} \\ \sin 3\degree+\sin 37\degree=2\sin \Big{(}(40\degree)/(2)\Big{)}\cos \Big((-34\degree)/(2)\Big) \\ \sin 3\degree+\sin 37\degree=2\sin (20\degree)\cos (-17\degree) \end{gathered}`