Question

Use the product-to-sum identities to rewrite the following expression as a sum or difference.Зл3sin () cos (35)5л2COS2

Answer 1

The expression is given below as

`3\sin ((5\pi)/(2))\cos ((3\pi)/(2))`

Concept:

The product to sum identity to be used is given below as

`\begin{gathered} \sin A\cos B=(1)/(2)(\sin (A+B)+\sin (A-B) \\ A=(5\pi)/(2),B=(3\pi)/(2) \end{gathered}`

By substituting the values, we will have

`\begin{gathered} 3\sin ((5\pi)/(2))\cos ((3\pi)/(2))=3*(1)/(2)(\sin ((5\pi)/(2)+(3\pi)/(2))+\sin ((5\pi)/(2)-(3\pi)/(2)) \\ 3\sin ((5\pi)/(2))\cos ((3\pi)/(2))=(3)/(2)(\sin ((5\pi+3\pi)/(2))+\sin ((5\pi-3\pi)/(2)) \\ 3\sin ((5\pi)/(2))\cos ((3\pi)/(2))=(3)/(2)(\sin ((8\pi)/(2))+\sin ((2\pi)/(2)) \\ 3\sin ((5\pi)/(2))\cos ((3\pi)/(2))=(3)/(2)(\sin ((8\pi)/(2))+\sin ((2\pi)/(2)) \\ 3\sin ((5\pi)/(2))\cos ((3\pi)/(2))=(3)/(2)(\sin (4\pi)+\sin (\pi) \end{gathered}`

Hence,

The final answer is

`\Rightarrow(3)/(2)(\sin (4\pi)+\sin (\pi)`