Complementary Angles Calculator
Solve complementary angles problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Complement = 90 - angle (degrees) or pi/2 - angle (radians)
Two angles are complementary when they sum to exactly 90 degrees (pi/2 radians). The complement of angle A is simply 90 - A. This relationship leads to the cofunction identities: sin(A) = cos(90-A), cos(A) = sin(90-A), tan(A) = cot(90-A).
Worked Examples
Example 1: Finding the Complement of 35 Degrees
Problem:Find the complementary angle of 35 degrees and verify the trigonometric cofunction identity.
Solution:Complement = 90 - 35 = 55 degrees\nVerification: sin(35) = 0.5736\ncos(55) = cos(90 - 35) = 0.5736\nsin(35) = cos(55), confirming the cofunction identity.\nAlso: cos(35) = 0.8192 = sin(55).
Result:The complement of 35 degrees is 55 degrees. sin(35) = cos(55) = 0.5736.
Example 2: Complementary Angles in a Right Triangle
Problem:In a right triangle with one acute angle of 72 degrees, find the other acute angle and compute trig ratios.
Solution:The other acute angle = 90 - 72 = 18 degrees\nsin(72) = 0.9511 = cos(18)\ncos(72) = 0.3090 = sin(18)\ntan(72) = 3.0777 and tan(18) = 0.3249\nNote: tan(72) * tan(18) = 1 (cofunctions of tangent multiply to 1).
Result:The complementary angle is 18 degrees. All cofunction identities are verified.
Frequently Asked Questions
What are complementary angles and how are they defined?
Complementary angles are two angles whose measures add up to exactly 90 degrees (or pi/2 radians). Each angle is called the complement of the other. For example, 30 degrees and 60 degrees are complementary because 30 + 60 = 90. Similarly, 45 degrees and 45 degrees are complementary. The angles do not need to be adjacent or share a common side; they simply need to sum to 90 degrees. Complementary angles are always acute (between 0 and 90 degrees), since if either angle were 90 or more degrees, their sum would exceed 90 degrees. This concept is fundamental in geometry, trigonometry, and many practical applications.
What is the difference between complementary and supplementary angles?
Complementary angles sum to 90 degrees, while supplementary angles sum to 180 degrees. The complement of a 30-degree angle is 60 degrees, but the supplement of 30 degrees is 150 degrees. Complementary angles are always acute (less than 90 degrees each), whereas supplementary angles can include one obtuse angle paired with an acute angle, or two right angles. A memory trick: 'C' comes before 'S' in the alphabet, and 90 comes before 180. Another mnemonic: complementary angles form a Corner (right angle), while supplementary angles form a Straight line. Both concepts appear extensively in geometry proofs and angle-chasing problems involving parallel lines.
How do complementary angles relate to trigonometric functions?
One of the most important trigonometric identities involves complementary angles: sin(A) = cos(90 - A) and cos(A) = sin(90 - A). This is called the cofunction identity, and it explains why cosine is named 'co-sine' meaning the sine of the complement. Similarly, tangent and cotangent are cofunctions: tan(A) = cot(90 - A). Secant and cosecant follow the same pattern: sec(A) = csc(90 - A). These identities are not mere coincidences but follow directly from the geometry of right triangles, where the opposite side of one acute angle is the adjacent side of the other acute angle. Understanding cofunction identities simplifies many trigonometric calculations and proofs.
Where do complementary angles appear in right triangles?
In every right triangle, the two acute angles are complementary because the three interior angles must sum to 180 degrees, and one angle is already 90 degrees, leaving exactly 90 degrees for the other two. This means if one acute angle is alpha, the other is (90 - alpha). This relationship is the geometric basis for cofunction identities: the opposite side of angle alpha is the adjacent side of angle (90 - alpha), so sin(alpha) = cos(90 - alpha). Knowing one acute angle immediately determines the other, which simplifies solving right triangles. Architects, surveyors, and engineers regularly use this property when working with right-angle constructions.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy