Finding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles

Using the formula for the cosine of the difference of two angles, find the exact value of $\cos (\frac{5 π}{4} - \frac{π}{6}) .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Begin by writing the formula for the cosine of the difference of two angles. Then substitute the given values.

$\begin{matrix} \cos (α - β) & = & \cos α \cos β + \sin α \sin β \\ \cos (\frac{5 π}{4} - \frac{π}{6}) & = & \cos (\frac{5 π}{4}) \cos (\frac{π}{6}) + \sin (\frac{5 π}{4}) \sin (\frac{π}{6}) \\ = & (- \frac{\sqrt{2}}{2}) (\frac{\sqrt{3}}{2}) - (\frac{\sqrt{2}}{2}) (\frac{1}{2}) \\ = & - \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \\ = & \frac{- \sqrt{6} - \sqrt{2}}{4} \end{matrix}$
Keep in mind that we can always check the answer using a graphing calculator in radian mode.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find the exact value of $\cos (\frac{π}{3} - \frac{π}{4}) .$

How did it go?