Cofunction Calculator - Complementary Trig Identities

A cofunction calculator is a trigonometry tool that evaluates the six cofunction identity pairs at the complementary angle for any input angle. Two angles are complementary when they sum to 90 degrees (pi/2 radians), and cofunction identities say that the value of one trigonometric function at an angle equals the value of its partner function at the complementary angle. The six identities are sin(θ) = cos(pi/2 - θ), cos(θ) = sin(pi/2 - θ), tan(θ) = cot(pi/2 - θ), cot(θ) = tan(pi/2 - θ), sec(θ) = csc(pi/2 - θ), and csc(θ) = sec(pi/2 - θ). The calculator returns all six at once.

• • Precalculus homework: Confirm cofunction identities like sin(30°) = cos(60°) = 0.5 while working through exercises.
• • Right triangle work: Translate a side ratio and its complement without redoing the same conversion twice.
• • Complementary-angle proofs: Use the six-pair table as a sanity check when walking through a complementary-angle derivation.
• • Physics and engineering identities: Resolve terms in wave, oscillator, and vector formulas that rely on the sin/cos, tan/cot, or sec/csc cofunction pairs.

Cofunction identities are sometimes confused with the reciprocal identities (csc = 1/sin, sec = 1/cos, cot = 1/tan). The cofunction identities pair sin with cos, tan with cot, and sec with csc, and they say those pairs take the same value at complementary angles.

Cofunction relationships show up the moment a problem involves a right triangle, because the two acute angles in a right triangle always sum to 90 degrees. The identity sin(θ) = cos(90 - θ) is what lets you swap sine and cosine freely inside a right triangle.

When the work shifts to the cosine side of the unit circle and you only have a cosine value to start from, the Arccos Calculator handles the inverse step so the cofunction pair still has a meaningful starting point.

The calculator reads the input angle and its unit, converts the angle to radians, computes the complementary angle as pi/2 minus the input in radians, and evaluates the six trig functions at both angles.

• θ: The original angle you enter. The calculator accepts either degrees or radians and converts internally to radians for evaluation.
• π/2 − θ: The complementary angle. In degrees it is 90 - θ, and in radians it is π/2 - θ. The cofunction identity compares a function at θ to its partner at this complementary angle.
• Pair (f, g): One of the three cofunction pairs: (sin, cos), (tan, cot), or (sec, csc). The identity f(θ) = g(π/2 - θ) holds for all three.

For the reciprocal partners (cot, sec, csc), the calculator returns the value at the original angle θ. The cofunction identity holds because the reciprocal of cot(π/2 − θ) is tan(π/2 − θ) = cot(θ), so cot(θ) and cot(π/2 − θ) are reciprocals across complementary angles.

According to Wolfram MathWorld: Complementary Angles, two angles are complementary when they sum to π/2 radians (90 degrees), and the function pairs (sin, cos), (tan, cot), and (sec, csc) take the same value at complementary angles.

If you only have a sine value and need the angle back, the Arcsin Calculator finishes the inverse step so the cofunction identity still has a meaningful starting point.

These four concepts come up every time you apply a cofunction identity.

The three pairs (sin, cos), (tan, cot), and (sec, csc) cover every common function. The result panel is built around those three families, so each pair gets its own row.

Negative angles and angles past 90° give cofunction values with the same sign as the original, because sine and cosine are odd and even around 0 respectively. When the input is exactly 0° or 90°, the reciprocal functions (cot, sec, csc) are undefined.

If you need the same angle reported in degrees, radians, and gradians while you work through the cofunction pairs, the Angle Converter keeps the units aligned without re-entering the input.

Working with the calculator takes a few seconds. Enter the angle, choose its unit, and read the complementary angle plus the six identity pair values in the result panel.

1. 1 Enter the angle: Type the angle into the input box. Positive, negative, and large angles all work, because the identity uses the value of the function at the angle, not the angle itself.
2. 2 Choose the angle unit: Pick degrees or radians from the unit selector. The complementary angle is reported in the same unit, so the result panel never mixes units.
3. 3 Read the complementary angle: The primary result is 90° − θ in degrees, or π/2 − θ in radians. This is the angle every cofunction pair compares against.
4. 4 Read the six identity pairs: Each row below shows one cofunction pair. The number reported is the shared value at θ and π/2 − θ, so you can confirm the identity at a glance.
5. 5 Watch for undefined values: When the input is 0° or 90° (or any multiple that zeros a denominator), the reciprocal functions sec, csc, cot, and tan become undefined. The result panel shows 'Undefined' for those rows.
6. 6 Reset for a new angle: Press Reset to clear the input and return to the default 30° starting point, which is the most common reference value for the identity.

A right triangle has one acute angle of 35° and another acute angle of 55°. Enter 35° in the input box, choose degrees, and read the complementary angle 55°. The sin/cos row reports sin(35°) = cos(55°) ≈ 0.5736, the cos/sin row reports cos(35°) = sin(55°) ≈ 0.8192, and the tan/cot row reports 0.7002.

If the angle came from a real right triangle and you also want the side lengths and the remaining angles, the Right Triangle Calculator carries the complementary-angle step into the rest of the triangle.

A cofunction calculator that returns all six identity pairs and the complementary angle in a single pass saves time on homework, derivations, and engineering checks.

• • All six pairs at once: See sin, cos, tan, cot, sec, and csc cofunction pair values side by side.
• • Complementary angle included: The complementary angle is reported in the input unit.
• • Degrees and radians in one input: A unit selector keeps the same calculation valid in either unit, with no need to multiply or divide by 180/π by hand.
• • Undefined values are explicit: When cot, sec, csc, or tan blow up at 0° or 90°, the result panel shows 'Undefined' instead of a confusing infinity.
• • Right-triangle friendly: The complementary angle for a right triangle is the other acute angle, so the result panel doubles as a quick check on a 30-60-90 or 45-45-90 triangle.
• • Pinned to authoritative sources: The cofunction definition and complementary-angle rule cite Wolfram MathWorld and Wikipedia.

The result panel shows the complementary angle as the primary output, then lists the six identity pairs as a single value each. If sin(θ) is 0.5 and the complementary angle is 60°, the row 'sin(θ) = cos(π/2 − θ)' reports 0.5.

For the reciprocal partners (cot, sec, csc), the calculator returns the value at the original angle θ. The cofunction identity holds because cot(π/2 - θ) = tan(θ) by the identity.

If you need the same cofunction pair in both units, the Radians to Degrees Calculator swaps degrees and radians without changing the value at the angle itself.

A few factors control what the calculator can give you. Knowing them up front prevents the most common mistakes.

• • The calculator reports the principal real value for each function. It does not evaluate cofunction identities in the complex plane, which is rarely what classroom or applied work requires.
• • The cofunction pair is shown as a single value, not as two side-by-side evaluations. Compute the complementary function by hand using the angles reported in the result panel.

If you need the complementary function written out, the result panel gives you the shared value of each pair. With θ = 30°, the cos/sin row reports 0.8660, which is the value of cos(30°) and also sin(60°). The row label makes that explicit, and the complementary angle 60° is the other side of the identity.

According to Wikipedia: Trigonometric functions, two angles are complementary when they sum to π/2 radians (90 degrees), and the function pairs (sin, cos), (tan, cot), and (sec, csc) take the same value at complementary angles.

If the problem hands you a tangent value and you need the angle back, the Arctan Calculator runs the same inverse workflow on the tangent side of the (tan, cot) cofunction pair.