Question

in the following problem, the expression is the right side of the formula for cos(α - β) with particular values for α and β. cos(83°)cos(38°)+sin(83°)sin(38°)
a. identify α and β in each expression.
the value for α: 83°
the value for β: 38°
b. write the expression as the cosine of an angle. cos
c. find the exact value of the expression.
(type an exact answer, using fraction, radicals and a rationalized denominator.)

Explanation:

Step1: Recall cosine - difference formula

The formula for $\cos(\alpha-\beta)=\cos\alpha\cos\beta + \sin\alpha\sin\beta$. Given $\cos(83^{\circ})\cos(38^{\circ})+\sin(83^{\circ})\sin(38^{\circ})$, we know from part (a) that $\alpha = 83^{\circ}$ and $\beta=38^{\circ}$.

Step2: Rewrite as cosine of an angle

Substitute $\alpha$ and $\beta$ into the formula. So, $\cos(83^{\circ})\cos(38^{\circ})+\sin(83^{\circ})\sin(38^{\circ})=\cos(83^{\circ}- 38^{\circ})=\cos(45^{\circ})$.

Step3: Find the exact value

We know that $\cos(45^{\circ})=\frac{\sqrt{2}}{2}$.

Answer:

b. $\cos(45^{\circ})$
c. $\frac{\sqrt{2}}{2}$