Complementary Angles Calculator

Find the complementary angle. Two angles are complementary when they add up to 90 degrees.

Enter an Angle

Angle must be between 0 and 90 degrees for a complementary angle to exist.

Result

Complementary Angle
55
degrees
Given Angle 35°
Complement 55°
Sum 90°
Given in Radians 0.6109 rad
Complement in Radians 0.9599 rad

Step-by-Step Solution

Complement = 90° - 35° = 55°

What Are Complementary Angles?

Two angles are called complementary angles when the sum of their measures equals exactly 90 degrees (a right angle). Each angle is said to be the "complement" of the other. This is one of the fundamental angle relationships in geometry and trigonometry.

For example, 30° and 60° are complementary because 30° + 60° = 90°. Similarly, 45° and 45° are complementary since they also sum to 90°. The concept applies whether the angles are adjacent (sharing a common side) or non-adjacent (located separately).

Complementary Angle Formula

In Degrees

Subtract the given angle from 90° to find its complement.

Complement = 90° - angle

In Radians

Subtract the given angle from pi/2 radians to find its complement.

Complement = pi/2 - angle

Valid Range

Both angles must be positive. The given angle must be between 0° and 90° (exclusive).

0° < angle < 90°

Trigonometric Identity

The sine of an angle equals the cosine of its complement, and vice versa.

sin(A) = cos(90° - A)

Complementary vs. Supplementary Angles

It is important not to confuse complementary angles with supplementary angles. While complementary angles sum to 90°, supplementary angles sum to 180°. A helpful memory trick: "C" comes before "S" in the alphabet, and 90 comes before 180 in counting.

Properties of Complementary Angles

  • Both angles must be acute (between 0° and 90°).
  • If two complementary angles are equal, each measures exactly 45°.
  • A right angle (90°) cannot have a complement since its complement would be 0°.
  • Complementary angles do not have to be adjacent or next to each other.
  • In a right triangle, the two non-right angles are always complementary.

Complementary Angles in Trigonometry

The relationship between complementary angles is deeply embedded in trigonometry. The "co-" prefix in cosine, cotangent, and cosecant literally means "complement." This gives rise to the cofunction identities:

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

Real-World Examples

Complementary angles appear frequently in everyday life and professional fields. In architecture, roof slopes often involve complementary angles. In navigation, course corrections use complementary angle calculations. In sports like billiards and basketball, understanding angle relationships helps predict ball trajectories.

Common Complementary Angle Pairs

  • 10° and 80°
  • 20° and 70°
  • 30° and 60° (common in equilateral triangle halves)
  • 45° and 45° (isosceles right triangle)
  • 15° and 75°
  • 1° and 89°