Problem 82 - Algebra and Trigonometry

Question

Use words to describe the formula for each of the following: the cosine of the sum of two angles.

Step-by-Step Solution

Answer

The formula for the cosine of the sum of two angles is \( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \), where \( \cos(A + B) \) represents the cosine of the sum of two angles A and B, and the right side illustrates the cosine of A times the cosine of B subtracted by the sine of A times the sine of B.

Step 1: Identify the formula

The first step is to identify the correct formula. The formula for the cosine of the sum of two angles (denoted as \( \cos(A + B) \)) is \( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \).

Step 2: Break down the formula

The second step is to break down different parts of the formula. The term \( \cos(A + B) \) represents the cosine of the sum of two angles A and B. The right side of the equation \( \cos(A)\cos(B) - \sin(A)\sin(B) \) includes two parts - the cosine of A times the cosine of B ( \( \cos(A)\cos(B) \)) and minus the sine of A times the sine of B ( \( \sin(A)\sin(B) \)).

Step 3: Explain the significance of the formula

This formula is significant as it allows us to convert the cosine of a sum of two angles into a simple form involving the cosines and sines of the original angles, making computations easier in various mathematical and physical contexts.