Question

Write the product as a sum or difference of trigonometric functions.10 sin 49° cos 111°

Answer 1

`5(\sin \text{ 160}\degree\text{ - sin 62}\degree)`

Explanation:

Given information

`10\text{ sin 49}\degree\cos \text{ 111}\degree`
`\text{Let }\alpha\text{ =49}\degree\text{ and }\beta\text{ = 111}\degree`

Recall that, the product as a sum containing only sine and cosine can be written below as

`sin\alpha\cos \beta\text{ = }(1)/(2)\lbrack\sin (\alpha\text{ + }\beta)\text{ + (sin }\alpha\text{ - }\beta)\rbrack`

The next step is to substitute the given value into the above formula

`\begin{gathered} 10\sin \text{ 49}\degree\cos 111\degree\text{ = 10 }*(1)/(2)\lbrack\sin (49\text{ + 111) + sin(}49\text{ - 111)} \\ 10\sin 49\degree\cos 111\degree\text{ = 5\lbrack{}sin(160}\degree)\text{ + sin (-62}\degree) \\ 10\text{ sin49}\degree\cos \text{ 111}\degree\text{ = 5 (sin 160}\degree\text{ - sin 62}\degree) \end{gathered}`