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Chapter 5: Problem 83

Use the product-to-sum formulas to write the product as a sum or difference. $$10 \cos 75^{\circ} \cos 15^{\circ}$$

Short Answer

Expert verified
The solution to the given problem is \(2.5\).

Step by step solution

01

Identify the given values

The given values are \(A = 75掳\) and \(B = 15掳\) in the product \(\cos A \cos B\) in the form \(10 \cos 75^{\circ} \cos 15^{\circ}\).
02

Insert the values into the product-to-sum formula

Fill the values of A and B into the formula \(\cos A \cos B=\frac{1}{2}[\cos(A-B)+\cos(A+B)]\), which becomes \(\frac{1}{2}[\cos(75掳-15掳)+\cos(75掳+15掳)]\).
03

Simplify the equation obtained

Calculate the values inside the brackets, which results in \(\frac{1}{2}[\cos 60^{\circ}+\cos 90^{\circ}]\).
04

Calculate the cosine of the angles

The cosine of 60掳, \(\cos 60掳= \frac{1}{2}\), and the cosine of 90掳, \(\cos 90掳=0\). Substitute these into the equation.
05

Final step

Multiply the obtained sum by 10 (as stated in the original product) to reach the final answer, which is \[10*\frac{1}{2}[\frac{1}{2}+0]\]

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