Cofunction Calculator
A cofunction calculator applies the six trigonometric cofunction identities, which say every trig function of an angle equals its cofunction evaluated at the complementary angle. Enter an angle in degrees or radians, pick a function, and get the matching cofunction expression, the exact value where it exists, and the step-by-step derivation.
Formula
Worked example
For sin(30 deg) in degrees: complement = 90 - 30 = 60 deg, so sin(30 deg) = cos(60 deg) = 1/2. In radians: sin(pi/6) = cos(pi/3) = 0.5.
What the cofunction identities actually say
Two trigonometric functions are cofunctions when one evaluated at an angle gives the same result as the other evaluated at the complementary angle. The complementary angle is whatever you must add to your original angle to reach a right angle, 90 degrees or pi/2 radians. Sine and cosine form one such pair, tangent and cotangent form another, and secant and cosecant form the third. The relationship comes directly from a right triangle: the sine of one acute angle is the opposite-over-hypotenuse ratio, but that same side is adjacent to the other acute angle, making it the cosine of that second angle. Because the two acute angles in any right triangle always sum to 90 degrees, the identity follows immediately.
Degrees versus radians: same identity, different notation
In degree mode the complementary angle is 90 deg - theta, so the identities read sin(theta) = cos(90 deg - theta) and so on. Switch to radians and 90 degrees becomes pi/2, so the same identities become sin(theta) = cos(pi/2 - theta). This calculator lets you toggle between the two units so you can work in whichever your textbook or exam uses. The decimal value that comes out is the same either way: sin(30 deg) and sin(pi/6) are identical, both equal to 0.5, because they describe the same geometric ratio.
Exact values at common angles
At multiples of 30 degrees (pi/6) and 45 degrees (pi/4) the trig functions take values that can be written as simple fractions involving square roots rather than infinite decimals. For example, sin(60 deg) = cos(30 deg) = sqrt(3)/2, which is exact, while the decimal 0.866025... is only an approximation. The "exact value" output and the common-angles table use these radical forms so you can copy the clean answer a teacher expects, not just a rounded decimal. For any other angle the exact form is not a simple radical, so the calculator falls back to a six-decimal approximation.
Using cofunction identities to simplify expressions
Cofunction identities are a practical algebraic tool. If you see sin(75 deg) and no angle-addition formula springs to mind, rewrite it as cos(15 deg) instead, which is easier to evaluate using cos(45 deg - 30 deg). In calculus, recognising that d/dx[sin x] = cos x is itself a cofunction relationship: differentiating shifts the function to its complement. In signal processing and Fourier analysis the identity shows up as the fact that a sine wave and a cosine wave are the same shape, just shifted by a quarter period. Every time you convert between a sine and a cosine form you are applying a cofunction identity, whether you name it that or not.
All six cofunction identities in degrees and radians
| Function of theta | Equals (degrees) | Equals (radians) |
|---|---|---|
| sin(theta) | cos(90 deg - theta) | cos(pi/2 - theta) |
| cos(theta) | sin(90 deg - theta) | sin(pi/2 - theta) |
| tan(theta) | cot(90 deg - theta) | cot(pi/2 - theta) |
| cot(theta) | tan(90 deg - theta) | tan(pi/2 - theta) |
| sec(theta) | csc(90 deg - theta) | csc(pi/2 - theta) |
| csc(theta) | sec(90 deg - theta) | sec(pi/2 - theta) |
Each function of theta equals its cofunction at the complementary angle.
Frequently asked questions
What are the six cofunction identities?
They are: sin(theta) = cos(90 deg - theta), cos(theta) = sin(90 deg - theta), tan(theta) = cot(90 deg - theta), cot(theta) = tan(90 deg - theta), sec(theta) = csc(90 deg - theta), and csc(theta) = sec(90 deg - theta). In radians replace 90 deg with pi/2. Each identity follows from the fact that the two acute angles in a right triangle sum to 90 degrees.
How do I use this calculator in radians?
Switch the "Angle unit" toggle to Radians, then enter your angle in radians (for example, pi/6 is about 0.5236). The calculator will compute the complement as pi/2 minus your angle and apply the appropriate cofunction identity. All six functions and their cofunctions work the same way in radians.
Why does sin(30 deg) equal cos(60 deg)?
Because 30 deg and 60 deg are complementary (they sum to 90 deg). In a right triangle with a 30 deg angle, the side opposite the 30 deg angle is adjacent to the 60 deg angle. The sine of 30 deg is opposite/hypotenuse and the cosine of 60 deg is adjacent/hypotenuse, and those are the same side, so both equal exactly 1/2.
Do cofunction identities work for angles larger than 90 degrees?
Yes. The identities hold for any angle where both functions are defined, not just acute angles between 0 and 90 degrees. For theta = 120 deg the complement 90 deg - 120 deg = -30 deg is negative, and sin(120 deg) still equals cos(-30 deg). Both equal sqrt(3)/2. The right-triangle picture gives the intuition, but the algebraic identity extends to the full real line.
What is the difference between a cofunction and a reciprocal identity?
A cofunction identity pairs functions at complementary angles: sin(theta) = cos(90 deg - theta). A reciprocal identity pairs functions at the same angle: csc(theta) = 1/sin(theta). They are separate relationships. Sine and cosecant are reciprocals of each other, but sine and cosine are cofunctions of each other.
What does the "exact value" output show?
For angles that are multiples of 30 degrees (pi/6) or 45 degrees (pi/4), the trig functions have exact values expressible as simple fractions with square roots, for example sqrt(3)/2 or sqrt(2)/2. The exact value output shows this form instead of a rounded decimal. For all other angles the value is irrational and cannot be simplified, so only the decimal approximation is shown.