Question

Find the exact value of the expression. No decimal answers. Show all work.Hint: Use an identity to expand the expression.

Answer 1

Given the expression:

`\cos ((\pi)/(4)+(\pi)/(6))`

You can expand it by using the following Identity:

`\cos \mleft(A+B\mright)\equiv cos(A)cos(B)-sin(A)sin(B)`

You can identify that, in this case:

`\begin{gathered} A=(\pi)/(4) \\ \\ B=(\pi)/(6) \end{gathered}`

Then, you can expand it as follows:

`\cos ((\pi)/(4)+(\pi)/(6))=cos((\pi)/(4))cos((\pi)/(6))-sin((\pi)/(4))sin((\pi)/(6))`

By definition:

`\cos ((\pi)/(4))=\frac{\sqrt[]{2}}{2}`
`\cos ((\pi)/(6))=\frac{\sqrt[]{3}}{2}`
`\sin ((\pi)/(4))=\frac{\sqrt[]{2}}{2}`
`\sin ((\pi)/(6))=(1)/(2)`

Then, you can substitute values:

`=(\frac{\sqrt[]{2}}{2})(\frac{\sqrt[]{3}}{2})-(\frac{\sqrt[]{2}}{2})((1)/(2))`

Simplifying, you get:

`\begin{gathered} =(\frac{\sqrt[]{2}}{2})(\frac{\sqrt[]{3}}{2})-(\frac{\sqrt[]{2}}{2})((1)/(2)) \\ \\ =\frac{\sqrt[]{6}}{4}-\frac{\sqrt[]{2}}{4} \end{gathered}`
`=\frac{\sqrt[]{6}-\sqrt[]{2}}{4}`

Hence, the answer is:

`\frac{\sqrt[]{6}-\sqrt[]{2}}{4}`