Question

in the following problem, the expression is the right side of the formula for $\cos(\alpha - \beta)$ with particular values for $\alpha$ and $\beta$.
$\cos(62^{\circ})\cos(32^{\circ})+\sin(62^{\circ})\sin(32^{\circ})$

a. identify $\alpha$ and $\beta$ in each expression.
the value for $\alpha$: $62^{\circ}$
the value for $\beta$: $32^{\circ}$
b. write the expression as the cosine of an angle. $\cos 30^{\circ}$
c. find the exact value of the expression.
(type an exact answer, using fraction, radicals and a rationalized denominator.)

Explanation:

Step1: Use the cosine difference formula

The formula for \(\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\). From part (b), we know the expression is \(\cos(62^{\circ} - 32^{\circ})=\cos30^{\circ}\).

Step2: Find the value of \(\cos30^{\circ}\)

We know that \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\)

Answer:

\(\frac{\sqrt{3}}{2}\)