Question

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

Answer 1

Given

$\sin\frac{19\pi}{30}\cos\frac{\pi}{5}+\cos\frac{19\pi}{30}\sin\frac{\pi}{5}$

Find

Exact value

Explanation

Here we use the sum formula,

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

on comparing we get ,

$A=\frac{19\pi}{5},B=\frac{\pi}{5}$

so ,

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

Final Answer

Therefore, the exact value is 1/2