Cofunction Identities
Reference for the six cofunction identities: sin/cos, tan/cot, and sec/csc.
Includes proofs and applications for simplifying trigonometric expressions.
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
Key Notes
- Cofunction identities arise because in a right triangle, the two acute angles always sum to 90° — each is the other's complement by definition
- In radians: sin(θ) = cos(π/2 − θ); substitute π/2 wherever you see 90°
- The "co" prefix is a direct reference to complement: cosine is the cofunction of sine, cotangent of tangent, cosecant of secant
- These identities apply only to complementary angles — they do not generalize to supplementary (180°) or arbitrary angle relationships
Key Notes
- Core identities: sin θ = cos(90°−θ); cos θ = sin(90°−θ); tan θ = cot(90°−θ); csc θ = sec(90°−θ); sec θ = csc(90°−θ). Each trig function equals the co-function of its complementary angle (90°−θ, or π/2−θ in radians).
- Why "co-function": In a right triangle, the two acute angles are complementary (sum to 90°). The sine of one angle equals the cosine of the other — since each is opposite/hypotenuse for one angle and adjacent/hypotenuse for the other. The "co" in cosine literally means "complementary sine."
- Useful for simplification: An expression like sin(55°) can be rewritten as cos(35°). This simplifies problems where one form is easier to work with, or allows matching a table entry in a different column.
- In radians: sin θ = cos(π/2−θ). For example, sin(π/3) = cos(π/6), and both equal √3/2. The cofunction identities hold for all real values of θ, not just acute angles, because of the symmetry of the unit circle.
- Applications: Cofunction identities are used to simplify trigonometric expressions, prove other identities, evaluate trig functions without a calculator (by converting to a known angle), and solve equations where the same value appears in two different trig functions.