Question

Use a sum or difference formula to find the exact value of the following.+sin19π30cosπ5cos19π30sinπ5

Answer 1

Given

`\sin(19\pi)/(30)\cos(\pi)/(5)+\cos(19\pi)/(30)\sin(\pi)/(5)`

Find

Exact value

Step-by-step explanation

Here we use the sum formula,

`\sin(A+B)=\sin A\cos B+\cos A\sin B`

on comparing we get ,

`A=(19\pi)/(5),B=(\pi)/(5)`

so ,

`\begin{gathered} \sin(19\pi)/(30)\cos(\pi)/(5)+\cos(19\pi)/(30)\sin(\pi)/(5)=\sin((19\pi)/(30)+(\pi)/(5)) \\ \\ \\ \sin(19\pi)/(30)\cos(\pi)/(5)+\cos(19\pi)/(30)\sin(\pi)/(5)=\sin((25\pi)/(30)) \\ \\ \sin(19\pi)/(30)\cos(\pi)/(5)+\cos(19\pi)/(30)\sin(\pi)/(5)=\sin((5\pi)/(6))=\sin(\pi-(\pi)/(6)) \\ \\ \sin(19\pi)/(30)\cos(\pi)/(5)+\cos(19\pi)/(30)\sin(\pi)/(5)=\sin((\pi)/(6)) \\ \\ \sin(19\pi)/(30)\cos(\pi)/(5)+\cos(19\pi)/(30)\sin(\pi)/(5)=(1)/(2) \\ \end{gathered}`

Final Answer

Therefore, the exact value is 1/2