Cofunction Calculator

Calculate cofunction identities for all six trigonometric functions with step-by-step solutions.

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Result

Cofunction Identity
sin(30) = cos(60)
= 0.5

Step-by-Step Solution

sin(x) = cos(90 - x)

Understanding Cofunction Identities

Cofunction identities are fundamental relationships in trigonometry that connect pairs of trigonometric functions. Two trigonometric functions are cofunctions of each other if their values are equal when their arguments sum to 90 degrees (or pi/2 radians). The word "cofunction" itself means "complementary function," referring to complementary angles that add up to 90 degrees.

These identities arise from the geometry of right triangles. In a right triangle, the two acute angles are complementary (they sum to 90 degrees). The sine of one acute angle equals the cosine of the other, and this relationship extends to all trigonometric function pairs.

The Six Cofunction Identities

Sine and Cosine

Sine and cosine are cofunctions of each other.

sin(x) = cos(90 - x)

Cosine and Sine

Cosine equals the sine of the complementary angle.

cos(x) = sin(90 - x)

Tangent and Cotangent

Tangent and cotangent are cofunctions of each other.

tan(x) = cot(90 - x)

Cotangent and Tangent

Cotangent equals the tangent of the complementary angle.

cot(x) = tan(90 - x)

Secant and Cosecant

Secant and cosecant are cofunctions of each other.

sec(x) = csc(90 - x)

Cosecant and Secant

Cosecant equals the secant of the complementary angle.

csc(x) = sec(90 - x)

How Cofunction Identities Work

Right Triangle Derivation

Consider a right triangle with angles A, B, and 90 degrees. Since A + B = 90 degrees, we have B = 90 - A. The side opposite to angle A is the side adjacent to angle B. Therefore, sin(A) = opposite/hypotenuse = adjacent(for B)/hypotenuse = cos(B) = cos(90 - A).

In Radians

When working in radians, 90 degrees is replaced by pi/2. So the identities become: sin(x) = cos(pi/2 - x), tan(x) = cot(pi/2 - x), sec(x) = csc(pi/2 - x), and vice versa for each pair.

Applications of Cofunction Identities

  • Simplifying expressions: Cofunction identities allow you to rewrite trigonometric expressions in simpler forms.
  • Solving equations: These identities can be used to convert between different trig functions when solving equations.
  • Proving other identities: Cofunction identities serve as building blocks for proving more complex trigonometric identities.
  • Calculus: These identities are useful in integration and differentiation of trigonometric functions.
  • Navigation and surveying: Complementary angle relationships are used in real-world measurement and navigation problems.

Special Angle Values

Cofunction identities are particularly easy to verify with special angles. For example, sin(30) = 0.5 and cos(60) = 0.5, confirming that sin(30) = cos(90 - 30) = cos(60). Similarly, tan(45) = cot(45) = 1, which makes sense since 45 is its own complement (90 - 45 = 45).

Tips for Using Cofunction Identities

  • Remember that "co" in cosine, cotangent, and cosecant stands for "complementary."
  • The complement of angle x is always (90 - x) in degrees or (pi/2 - x) in radians.
  • Cofunction identities only directly apply to complementary angles (summing to 90 degrees).
  • These identities hold for all angle values, not just acute angles, due to the unit circle definition.