Problem 89 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The sine of an angle is equal to the cosine of its complement, and vice versa, this is known as the co-function identity. This relationship also holds true among other pairs of trigonometric functions.

Step 1: Understand the Concept of Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. In degrees, if two angles are complementary, their measures are \(x\) and \(90 - x\). In radians, these angles are often described as \(r\) and \(\frac{\pi}{2} - r\).

Step 2: Apply the Concept to Trigonometric Functions

For any acute angle \(x\), the sine of \(x\) (or \(\sin(x)\)) is equal to the cosine of its complementary angle (or \(\cos(90 - x)\)) in degrees. Similarly, the cosine of an angle is equal to the sine of its complement. This is known as co-function identity. It can be written in a general way as: \(\sin(x) = \cos(90 - x)\) and \(\cos(x) = \sin(90 - x)\). This relationship holds true for other trigonometric functions as well, including tangent and cotangent, secant and cosecant.

Step 3: Understand the Significance

These relationships among trigonometric functions dramatically simplify the process of solving trigonometric equations and problems. This understanding is fundamental in calculus, physics and engineering problem solving practices.