Cofunction Identities
Relate trig functions of complementary angles. sin(θ) = cos(90° - θ) and other cofunction pairs.
The Formulas
sin(θ) = cos(90° - θ)
cos(θ) = sin(90° - θ)
tan(θ) = cot(90° - θ)
cot(θ) = tan(90° - θ)
sec(θ) = csc(90° - θ)
csc(θ) = sec(90° - θ)
cos(θ) = sin(90° - θ)
tan(θ) = cot(90° - θ)
cot(θ) = tan(90° - θ)
sec(θ) = csc(90° - θ)
csc(θ) = sec(90° - θ)
Cofunction identities show that any trig function of an angle equals its cofunction of the complement. Two angles are complementary when they add up to 90° (or π/2 radians).
Variables
| Symbol | Meaning |
|---|---|
| θ | Any angle |
| 90° - θ | The complementary angle |
Example 1
Verify that sin(40°) = cos(50°)
40° + 50° = 90° (they are complementary)
sin(40°) ≈ 0.6428
cos(50°) ≈ 0.6428
They are equal, confirming the cofunction identity.
Example 2
Simplify sin(20°) × sec(70°)
sec(70°) = 1/cos(70°)
cos(70°) = sin(90° - 70°) = sin(20°)
So sec(70°) = 1/sin(20°)
sin(20°) × (1/sin(20°)) = 1
When to Use Them
Use cofunction identities when:
- Simplifying expressions involving complementary angles
- Proving other trigonometric identities
- Solving problems in right triangle geometry
- Converting between sine and cosine (or other cofunctions) for easier computation