Problem 86 - Algebra and Trigonometry

Question

Use words to describe the formula for: the power-reducing formula for the cosine squared of an angle.

Step-by-Step Solution

Answer

The power-reducing formula for the cosine squared of an angle is given by \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). This formula means that the square of the cosine of any angle is equal to the average of 1 and the cosine of twice that angle.

Step 1: Identify the original formula

The power-reducing formula for the cosine squared of an angle is \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). This formula represents a relationship between the square of the cosine of an angle and the cosine of twice that angle.

Step 2: Break down the left-hand side of the formula

The term \( \cos^2(x) \) is the square of the cosine of an angle. Squaring a number (or function) is equivalent to multiplying that number (or function) by itself. Here, \( \cos^2(x) \) means 'the cosine of x multiplied by the cosine of x'.

Step 3: Break down the right-hand side of the formula

On the right-hand side of the equation, \( \cos(2x) \) represents the cosine of an angle that is twice as large as x. The '+' operator adds this to 1, and the entire sum is divided by 2 which averages the value.

Step 4: Interpret the entire formula

The formula \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \) shows that the square of the cosine of an angle can be calculated as the average of 1 and the cosine of twice that angle. This transformation can simplify computations in various contexts, particularly when integrating certain trigonometric functions.