Cofunction Calculator
The cofunction identities are among the most elegant relationships in trigonometry. They state that every trigonometric function of an angle is equal to its paired cofunction evaluated at the complementary angle, which is 90 degrees minus the original angle. The three cofunction pairs are sine and cosine, tangent and cotangent, and secant and cosecant. The prefix co- in cosine, cotangent and cosecant literally refers to complement: cosine is the sine of the complementary angle. So if you know sin(30) = 0.5, you automatically know that cos(60) = 0.5 without any extra work, because 30 and 60 are complementary. The same applies to all six trig functions. This is useful in simplifying expressions, proving trig identities, and working with right-angled triangles where the two acute angles always add up to 90 degrees. This calculator takes any angle in degrees and returns all six trig values at that angle and all six at the complementary angle. It is set up so you can verify the cofunction identity visually: each value in the left column should match its cofunction in the right column. The tool handles angles outside 0 to 90 degrees too, where the cofunction identity still holds algebraically. If a function is undefined at your chosen angle (tangent at 90 degrees, for example), the calculator shows that clearly.
Cofunction identities: sin(t)=cos(90-t), cos(t)=sin(90-t), tan(t)=cot(90-t), cot(t)=tan(90-t), sec(t)=csc(90-t), csc(t)=sec(90-t).
| Function | At θ | Cofunction | At (90 - θ) |
|---|---|---|---|
| sin(θ) | 0.5000 | cos(90-θ) | 0.5000 |
| cos(θ) | 0.8660 | sin(90-θ) | 0.8660 |
| tan(θ) | 0.5774 | cot(90-θ) | 0.5774 |
| cot(θ) | 1.7321 | tan(90-θ) | 1.7321 |
| sec(θ) | 1.1547 | csc(90-θ) | 1.1547 |
| csc(θ) | 2.0000 | sec(90-θ) | 2.0000 |
How it works
For any angle theta in degrees, the complement is (90 - theta). The cofunction identities are:
sin(theta) = cos(90 - theta)
cos(theta) = sin(90 - theta)
tan(theta) = cot(90 - theta)
cot(theta) = tan(90 - theta)
sec(theta) = csc(90 - theta)
csc(theta) = sec(90 - theta)
These hold for all angles where the functions are defined. In this calculator, angles are converted to radians before applying JavaScript's built-in Math functions. Cotangent, secant and cosecant are derived as 1/tan, 1/cos and 1/sin respectively.
Worked example
With angle = 30 degrees: complement = 60 degrees. sin(30) = 0.5000 and cos(60) = 0.5000, confirming the identity. cos(30) = 0.8660 and sin(60) = 0.8660. tan(30) = 0.5774, cot(30) = 1.7321, sec(30) = 1.1547, csc(30) = 2.0000. Each value at 30 degrees equals its cofunction pair at 60 degrees.
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