Overview
This lesson introduces cofunction identities in trigonometry, explains their origins using triangles, and demonstrates how to rewrite trigonometric functions in terms of their cofunctions with examples in both degrees and radians.
Triangle Angle Properties
- The sum of the interior angles of any triangle is 180° (π radians).
- In a right triangle, the two acute angles are complementary (sum to 90° or π/2 radians).
Cofunction Identities
- A function f is a cofunction of g if f(A) = g(B) when A and B are complementary angles (A + B = 90° or π/2).
- Common cofunction identities (in degrees):
- sin(A) = cos(90° − A)
- cos(A) = sin(90° − A)
- sec(A) = csc(90° − A)
- csc(A) = sec(90° − A)
- tan(A) = cot(90° − A)
- cot(A) = tan(90° − A)
- These identities also apply in radians using π/2 instead of 90°.
Examples in Degrees
- sin(18°) = cos(72°) because 18° + 72° = 90°.
- tan(65°) = cot(25°) because 65° + 25° = 90°.
- csc(84°) = sec(6°) because 84° + 6° = 90°.
- You can verify these using a calculator.
Examples in Radians
- cos(π/4) = sin(π/4) because π/4 + π/4 = π/2.
- cot(π/3) = tan(π/6) because π/3 + π/6 = π/2.
- sec(π/6) = csc(π/3) because π/6 + π/3 = π/2.
- All can be verified using the unit circle or calculator.
Key Terms & Definitions
- Complementary Angles — Two angles whose sum is 90° (π/2 radians).
- Cofunction — A pair of trigonometric functions related such that f(A) = g(90° − A) for complementary A and B.
Action Items / Next Steps
- Review the six right triangle trigonometric function definitions.
- Practice rewriting trigonometric expressions using cofunction identities.
- Verify cofunction identities for given angles with a calculator.
- Prepare for solving equations involving cofunction identities.