Cofunction Calculator

Apply the trigonometric cofunction identities — sin θ = cos(90° − θ), tan θ = cot(90° − θ), sec θ = csc(90° − θ) — to find a function's value and its complementary-angle match.

Quick Facts

Cofunction pairs
sin↔cos, tan↔cot, sec↔csc
Each pair matches at complementary angles: f(θ) = cofunction(90° − θ).
Why it works
Complementary angles sum to 90° (π/2 radians)
In a right triangle, the two acute angles are complementary, so swapping which one you measure from swaps opposite and adjacent sides.

Your Results

Calculated
Function Value
-
f(θ) at your angle
Complementary Angle
-
90° − θ (or π/2 − θ)
Cofunction Value
-
Same value, via the cofunction
Cofunction Pair
-
Function ↔ its cofunction

Ready

Enter an angle, choose a function, then press Calculate.

How the Cofunction Calculator works

Cofunction identities relate a trigonometric function of an angle to a different function evaluated at that angle's complement. Two angles are complementary when they add up to 90° (or π/2 radians). The three cofunction pairs are sine/cosine, tangent/cotangent, and secant/cosecant.

The formulas

For any angle θ:

  • sin(θ) = cos(90° − θ) and cos(θ) = sin(90° − θ)
  • tan(θ) = cot(90° − θ) and cot(θ) = tan(90° − θ)
  • sec(θ) = csc(90° − θ) and csc(θ) = sec(90° − θ)

This calculator takes the angle and function you choose, computes the function's value directly, finds the complementary angle (90° − θ, or π/2 − θ in radians), and evaluates the cofunction at that complementary angle to show that both sides of the identity agree.

Why the identities hold

In a right triangle, the two non-right angles always sum to 90°. The side that is "opposite" one of those angles is "adjacent" to the other. Since sine is opposite over hypotenuse and cosine is adjacent over hypotenuse, viewing the triangle from the complementary angle swaps which side is opposite and which is adjacent — turning a sine into a cosine, and vice versa. The same reasoning, applied to the other side ratios, gives the tan/cot and sec/csc pairs.

Where the functions are undefined

Tangent and secant divide by cosine, so they are undefined wherever cos(θ) = 0 — at 90°, 270°, and every 180° after that. Cotangent and cosecant divide by sine, so they are undefined wherever sin(θ) = 0 — at 0°, 180°, 360°, and so on. The calculator checks for these cases and flags them instead of returning an infinite result.

Reading the output

The "Function Value" and "Cofunction Value" results should always match (within floating-point rounding), because that agreement is exactly what the identity states. If you change the angle unit to radians, the calculator converts internally so the same identities still hold using π/2 in place of 90°.

Frequently Asked Questions

What is a cofunction identity?
A cofunction identity relates a trigonometric function of an angle to a different function of that angle's complement (90 degrees minus the angle). The pairs are sine and cosine, tangent and cotangent, and secant and cosecant: sin(θ) = cos(90°−θ), tan(θ) = cot(90°−θ), and sec(θ) = csc(90°−θ).
Why do sine and cosine of complementary angles match?
In a right triangle the two non-right angles always add up to 90 degrees. The side that is "opposite" one acute angle is "adjacent" to the other, so sine (opposite over hypotenuse) for one angle equals cosine (adjacent over hypotenuse) for its complement. The same swap of opposite and adjacent explains the tan/cot and sec/csc pairs.
Does the identity work in radians as well as degrees?
Yes. In radians the complementary angle is pi/2 minus the angle instead of 90 degrees minus the angle. Select Radians as the angle unit and the calculator converts automatically before applying the identity.
When are tangent, cotangent, secant, or cosecant undefined?
Tangent and secant are undefined wherever cosine is zero (90°, 270°, and every 180° from there). Cotangent and cosecant are undefined wherever sine is zero (0°, 180°, 360°, and so on). The calculator flags these angles instead of returning a result.

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