7.10: Trigonometric Identities III

Cofunction identities are a key concept in trigonometry. They describe how trigonometric functions relate when their input angles are complementary — meaning the angles add up to 90°. On the unit circle, every angle θ— measured counterclockwise from the positive x-axis — corresponds to a point with coordinates (cos⁡ θ, sin ⁡θ). These values represent the horizontal and vertical components of the terminal side of the angle.

If the same point on the unit circle is instead described using the complementary angle 90° − θ, the reference changes to the positive y-axis. In this case, the horizontal segment that was originally adjacent to θ becomes the side opposite to 90° - θ, while the vertical segment that was opposite becomes adjacent. Thus, the coordinates can equivalently be expressed as (sin⁡(90°− θ), cos⁡(90° − θ)). Since both coordinate descriptions refer to the same point, it follows that:

These equalities are the basic cofunction identities, which illustrate how sine and cosine interchange roles for complementary angles.

In addition to sine and cosine, the same reasoning applies to other trigonometric functions, giving the complete set of cofunction identities:

These identities demonstrate the symmetry of trigonometric functions and their deep connection to complementary angles in right triangles and the unit circle. They provide a systematic way to transform between functions when working with angles measured from different reference directions.