Problem 65 - Precalculus Essentials

Question

Describe a relationship among trigonometric functions that is based on angles that are complements.

Step-by-Step Solution

Answer

The relationship among trigonometric functions based on complementary angles is given by the identities \( \sin(90^\circ - A) = \cos A \) and \( \cos(90^\circ - A) = \sin A \) for any angle A. This is due to the sine of one angle being equal to the cosine of its complement and vice versa.

Step 1: Define Complementary Angles

Two angles are said to be complementary if the sum of their measures equals 90 degrees. We use this quality to show the relationship between sine and cosine.

Step 2: State the Relationship

The main relationship between trigonometric functions based on angles that are complements is given by the identities \( \sin(90^\circ - A) = \cos A \) and \( \cos(90^\circ - A) = \sin A \) for any angle A. This relationship is also valid in radians, where 90 degrees corresponds to \( \frac{\pi}{2} \) radians: \( \sin(\frac{\pi}{2} - A) = \cos A \) and \( \cos(\frac{\pi}{2} - A) = \sin A \).

Step 3: Explain the Relationship

If two angles are complementary, the sine of one angle will be equal to the cosine of the other, and vice versa. This is due to the rotational symmetry of the unit circle definition of sine and cosine, where the sine represents the y-coordinate and the cosine represents the x-coordinate. If we rotate by \( \frac{\pi}{2} \) (or 90 degrees) radians, the x and y coordinates are swapped, which gives us this relationship.