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Cofunction Calculator

Calculate, understand, and apply cofunction identities accurately for trigonometry, geometry, education, engineering, and angle transformations.

Learn what a cofunction is

Cofunction Calculator

Select a trigonometric function and angle in degrees to get the matching cofunction identity and numeric check.

Result

sin(x) = cos(90° - x), so sin(30°) = cos(60°)

Numeric check: 0.500000 ≈ 0.500000

What Is a Cofunction?

A cofunction is a trigonometric function paired with another function through a complementary angle relationship. When one angle is θ, the complementary angle is 90° - θ.

This pairing helps simplify expressions in right triangle trigonometry, unit circle work, and exam-style identity problems.

  • Definition: function pairs linked by 90° - θ
  • Meaning: complementary angle transformation
  • Common pairs: sin/cos, tan/cot, sec/csc
  • Used in geometry, physics, and engineering angle work

Cofunction Formula

Cofunction identities rewrite one trigonometric function as its partner evaluated at the complementary angle. The core identity is shown below.

Core identity: sin(90° - θ) = cos(θ)

Equivalent forms include cos(90° - θ) = sin(θ), tan(90° - θ) = cot(θ), and sec(90° - θ) = csc(θ).

In radians, replace 90° with π/2.

How to Calculate Cofunctions

This method works for homework, classroom exercises, and quick engineering checks where complementary angle symmetry is useful.

  1. Identify the given function (sin, cos, tan, cot, sec, or csc).
  2. Find the complementary angle: 90° - x.
  3. Replace the original function with its cofunction partner at the complementary angle.
  4. Use the calculator above to verify the identity with your angle.
  5. Simplify the final expression and check special values when possible.

Cofunction Examples

Try these values in the calculator to confirm each identity numerically.

  • sin(30°) = cos(60°)
  • cos(20°) = sin(70°)
  • tan(45°) = cot(45°)
  • sec(30°) = csc(60°)
  • sin(π/6) = cos(π/3) in radians
  • tan(π/4) = cot(π/4) in radians

Cofunction Identities

These identities are the foundation for simplifying trigonometric expressions before moving to advanced topics like double-angle or inverse functions.

  • sin(x) = cos(90° - x)
  • cos(x) = sin(90° - x)
  • tan(x) = cot(90° - x)
  • cot(x) = tan(90° - x)
  • sec(x) = csc(90° - x)
  • csc(x) = sec(90° - x)

Complementary Angles Calculator

Complementary angles add to 90°. In cofunction work, if your input angle is x, the paired angle is 90° - x.

Example: if x = 25°, the complementary angle is 65°. Then sin(25°) = cos(65°).

Use the calculator section above to transform any supported function with your angle input.

Cofunctions in Trigonometry

On the unit circle, cofunction pairs reflect complementary angle positions in the first quadrant. In right triangles, one acute angle and its complement connect opposite and adjacent ratios.

Cofunctions also support graph reasoning: sine and cosine curves are phase-shift partners under complementary angle substitution.

  • Unit circle interpretation
  • Right triangle ratio relationships
  • Expression simplification
  • Graph and symmetry reasoning

Cofunction vs Reciprocal Functions

Cofunction identities change the function type using 90° - x. Reciprocal identities change function type using division, such as sec(x) = 1/cos(x).

Students often mix these rules. Keep them separate: cofunction = complementary angle partner, reciprocal = multiplicative inverse.

Work through side-by-side examples to compare cofunction swaps with reciprocal inverses.

Cofunction Calculator

The interactive tool is placed above this section so you can calculate first, then read supporting theory.

Enter a function and angle in degrees, then review the identity form and numeric check output.

Jump back to the calculator to run another example.

Common Cofunction Mistakes

Check each step by writing the full identity before substituting numbers.

  • Using 180° - x instead of 90° - x
  • Pairing sin with tan instead of sin with cos
  • Forgetting to switch function type and angle together
  • Mixing cofunction rules with reciprocal identities
  • Dropping degree/radian mode consistency

Cofunction Identities on the Unit Circle

For acute angles in Quadrant I, complementary positions preserve ratio values while swapping function roles. This is why sin(θ) and cos(90° - θ) produce equal outputs.

Understanding this geometry makes identity memorization easier and supports proof writing in trigonometry courses.

FAQs About Cofunctions