Cofunction Calculator
Calculate trigonometric functions and their complementary cofunction relationships
Enter an angle and select a trigonometric function to discover the cofunction identity.
Cofunction Calculator
Calculate trigonometric functions and their complementary cofunction relationships
About the Cofunction Calculator
Cofunction identities are one of the cleanest bridges between geometry and algebra in trigonometry. They say that certain trigonometric functions become each other when the angle is replaced by its complement. In plain language, the sine of an angle equals the cosine of the angle needed to complete a right angle, the tangent of an angle equals the cotangent of its complement, and the secant of an angle equals the cosecant of its complement. Written symbolically, sin(θ) = cos(90° − θ), tan(θ) = cot(90° − θ), and sec(θ) = csc(90° − θ), with the same idea holding in radians as π/2 − θ. This calculator automates those identities so you can check values instantly and see the matching cofunction side by side.
The reason the identities work comes directly from right-triangle structure. In a right triangle, the two acute angles always add to 90°. The side opposite one acute angle is the side adjacent to the other, so the ratio definitions swap roles. Because sine is opposite over hypotenuse and cosine is adjacent over hypotenuse, they match across complementary angles. The same swap explains tangent and cotangent, as well as secant and cosecant. Once you see that side relationships are mirrored, the cofunction pattern becomes natural rather than mysterious.
This tool lets you work in either degrees or radians, which is important because trigonometry appears in school math, engineering, physics, graphics, navigation, and signal processing. In degree mode, the complementary angle is computed as 90° minus the entered angle. In radian mode, it is computed as π/2 minus the entered angle. The calculator evaluates the original function and the paired cofunction numerically so you can compare them directly. Small differences in displayed decimals can come from floating-point rounding, but mathematically the identity is exact wherever both expressions are defined.
Domain restrictions matter too. Tangent and secant are undefined whenever cosine is zero, which happens at 90° + n·180° or π/2 + nπ. Cotangent and cosecant are undefined whenever sine is zero, which happens at n·180° or nπ. The calculator checks those cases and warns you instead of displaying misleading values. Use it to verify homework, build intuition for complementary angles, prepare lessons, or quickly confirm trig identities during technical work.
Cofunction Examples
These worked examples show how the same value appears again when you switch to the matching cofunction of the complementary angle.
| Function & Angle | Result | Cofunction Identity |
|---|---|---|
| sin with 30° | sin(30°) = 0.5 | cos(60°) = 0.5, so the sine value matches the cosine of the complementary angle. |
| tan with 45° | tan(45°) = 1 | cot(45°) = 1 because 45° is its own complement inside a right angle pair. |
| sec with 60° | sec(60°) = 2 | csc(30°) = 2, illustrating the secant–cosecant cofunction relationship. |
| cos with 1.047 rad | cos(1.047 rad) ≈ 0.5001710746 | sin(0.5237963268 rad) ≈ 0.5001710746 because π/2 − 1.047 rad is about 0.5237963268 rad. |
How to Use
- Choose the trigonometric function you want to evaluate: sin, cos, tan, cot, sec, or csc.
- Select whether your angle is measured in degrees or radians, then enter the angle value in the input field.
- Click Calculate Cofunction to compute the original value, the complementary angle, and the paired cofunction value.
- Compare the displayed identity to see how the original function matches the cofunction of the complementary angle.
- Use Reset Calculator to clear the inputs and start a new trig example.
FAQ
What is a cofunction identity?
A cofunction identity states that a trigonometric function of an angle equals a paired trigonometric function of the angle's complement. The standard pairs are sin and cos, tan and cot, and sec and csc.
Why do complementary angles matter in trigonometry?
In any right triangle, the two acute angles add to 90°. That makes the opposite side for one angle become the adjacent side for the other, which is why sine turns into cosine and tangent turns into cotangent.
Can I use radians instead of degrees?
Yes. In radian mode the calculator uses π/2 minus the entered angle to find the complement, then evaluates the matching cofunction at that radian measure.
Why does the calculator sometimes say the function is undefined?
Some trigonometric functions divide by sine or cosine. If the denominator becomes zero, the function does not exist at that angle. For example, sec and tan are undefined where cosine is zero.
Should the original value and cofunction value always match exactly?
Mathematically yes, wherever the identity is defined. On a computer you may see tiny decimal differences far beyond the shown digits because calculations use floating-point arithmetic.