Complementary Angles Calculator
Enter one angle to find its complement (90 degrees minus the angle), or enter two angles to check whether they add up to 90 degrees. Switch between degrees and radians at any time. The trig cofunction table updates live so you can see how sine pairs with cosine, tangent with cotangent, and secant with cosecant for your exact pair of angles.
Formula
Worked example
An angle of 35 degrees: complement = 90 - 35 = 55 degrees. Check: 35 + 55 = 90. Trig cofunctions: sin(35) = cos(55) ≈ 0.5736, cos(35) = sin(55) ≈ 0.8192, tan(35) = cot(55) ≈ 0.7002.
What are complementary angles?
Two angles are complementary when they add up to exactly 90 degrees (pi/2 radians). They do not need to be next to each other: adjacent complementary angles share a common vertex and ray and together form a right angle, while non-adjacent complementary angles simply have a sum of 90 degrees regardless of where they appear. The two acute angles in any right triangle are always complementary, because the three angles must sum to 180 degrees and one of them is already 90 degrees.
How to find the complement of an angle
The formula is straightforward: complement = 90 degrees - angle. In radians, complement = pi/2 - angle. Every angle between 0 and 90 degrees (inclusive) has exactly one complementary angle. An angle of 0 degrees is complementary to 90 degrees, a 45-degree angle is complementary to itself, and no angle greater than 90 degrees has a complement (because the result would be negative). To check whether two given angles are complementary, add them and see whether the sum equals 90 degrees.
The cofunction identities and why they matter
The prefix "co-" in cosine, cotangent, and cosecant comes directly from "complement." For any pair of complementary angles A and B (A + B = 90 deg), the six trigonometric cofunction identities hold: sin(A) = cos(B), cos(A) = sin(B), tan(A) = cot(B), cot(A) = tan(B), sec(A) = csc(B), and csc(A) = sec(B). These relationships are used constantly in calculus, signal processing, and geometry. For example, when A = 30 deg and B = 60 deg: sin(30 deg) = 0.5 = cos(60 deg), and tan(30 deg) = 1/sqrt(3) = cot(60 deg). Knowing one side of each identity lets you derive the other without a separate calculation.
Complementary vs. supplementary angles
A common source of confusion is the difference between complementary and supplementary angles. Complementary pairs sum to 90 degrees, supplementary pairs sum to 180 degrees. A memory aid: "c" comes before "s" in the alphabet, and 90 comes before 180, so complementary goes with 90 and supplementary goes with 180. In geometry, supplementary angles form a straight line when placed adjacent to each other, while complementary angles form a right angle.
Common complementary angle pairs
| Angle A (deg) | Complement B (deg) | sin(A) = cos(B) | cos(A) = sin(B) | tan(A) = cot(B) |
|---|---|---|---|---|
| 0 | 90 | 0.000000 | 1.000000 | 0.000000 |
| 15 | 75 | 0.258819 | 0.965926 | 0.267949 |
| 30 | 60 | 0.500000 | 0.866025 | 0.577350 |
| 45 | 45 | 0.707107 | 0.707107 | 1.000000 |
| 60 | 30 | 0.866025 | 0.500000 | 1.732051 |
| 75 | 15 | 0.965926 | 0.258819 | 3.732051 |
| 90 | 0 | 1.000000 | 0.000000 | undefined |
Frequently encountered complementary pairs and their trig cofunction values.
Frequently asked questions
What does complementary mean in geometry?
Two angles are complementary when their measures add up to 90 degrees (a right angle). The word comes from the Latin "complementum," meaning something that completes. Complementary angles complete a right angle, just as supplementary angles (summing to 180 degrees) complete a straight line.
Can complementary angles be greater than 90 degrees?
No. Since the two angles must sum to exactly 90 degrees, neither angle can be greater than 90 degrees (unless you allow the other to be negative, which is not standard in basic geometry). Each complementary angle must be between 0 and 90 degrees, inclusive.
Do complementary angles have to be adjacent?
No. Two angles are complementary as long as their measures sum to 90 degrees, regardless of position. Adjacent complementary angles share a vertex and a side and together visually form a right angle, but non-adjacent complementary angles can appear anywhere - such as the two acute angles in a right triangle, which are complementary but sit at different vertices.
What is the complement of 45 degrees?
45 degrees. This is the only angle whose complement equals itself. It arises naturally in isosceles right triangles (45-45-90 triangles), where both acute angles are 45 degrees.
How do complementary angles relate to trigonometry?
Complementary angles are the basis of the cofunction identities: sin(A) = cos(90 - A), tan(A) = cot(90 - A), and sec(A) = csc(90 - A). That is exactly why cosine, cotangent, and cosecant have the prefix "co-" (short for complement). These identities let you convert between a trig function and its cofunction by replacing the angle with its complement, which is especially useful when angles are expressed as expressions rather than numbers.
How do I check if two angles are complementary?
Add the two angle measures. If the sum equals exactly 90 degrees (or pi/2 radians), they are complementary. Use the "Check if two angles are complementary" mode in this calculator to enter both angles and get an instant answer, including the exact sum and how far off 90 degrees the pair is if they do not qualify.