Question

What general conclusion can you draw about the sines and cosines of complementary angles? State your conclusion in words and in the form of equations.

Answer 1

The sines and cosines of complementary angles are equal to each other: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ), where θ and its complement add up to 90 degrees.

Step-by-step explanation:

The general conclusion that can be drawn about the sines and cosines of complementary angles is based on the trigonometric identity which states that the sine of an angle is equal to the cosine of its complement, and vice versa. In other words, sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ), where θ and 90° - θ are complementary angles (their sum equals 90 degrees).

For example, if one angle in a right triangle is 30 degrees, its complement is 60 degrees. Therefore, sin(30°) = cos(60°) and cos(30°) = sin(60°). This concept is a fundamental component of trigonometry and applies in various scenarios, including calculating the components of vectors and using trigonometric relationships such as the Pythagorean theorem in right angled triangles.