How this instrument works
A cofunction identity ties a trig function of an angle to a different trig function of its complement: sin θ = cos(90° − θ). The reason is a right triangle viewed from two seats. Sit at angle θ and one leg is 'opposite' you while the other is 'adjacent'. Sit at the other acute angle, 90° − θ, and those two legs swap roles — what was opposite is now adjacent, and vice versa — while the hypotenuse never moves. Swap the legs in the ratio and sine for one angle becomes cosine for the other.
The name 'cosine' is not decoration; it records this exact fact. Sine was already in use when the English mathematician Edmund Gunter coined 'cosine' and 'cotangent' in the 1620s, using the co- prefix to flag 'of the complement' throughout the function family — cosine, cotangent, cosecant each pair with sine, tangent, secant across the same 90° split. Every 'co' in trigonometry is a cofunction identity wearing a name tag.
The rule is defined over the full 0°-90° range but is worth watching at its middle and its ends. At θ = 45° the angle equals its own complement, so sine and cosine are forced to meet at one value — the symmetry behind an isosceles 45-45-90 triangle. At the edge θ = 0°, the complement swells to a full 90°, and sin 0° = 0 meets cos 90° = 0: the 'opposite' leg has shrunk to nothing, and both sides of the identity agree on that emptiness.
- Type your angle into Angle, θ — it reads degrees by default; switch its unit selector to radians or turns if that suits your work.
- Read Complement, 90° − θ for the partner angle that closes out the right angle alongside θ.
- Read sin(θ) = cos(complement) for the value both functions reach — sine of θ computed one way, cosine of the complement computed the other, always the same number.
- Keep θ between 0° and 90°; a cofunction pair only exists between two acute angles that sum to a right angle, and the sheet flags anything outside that band.
Worked example — a 30° angle and its 60° complement
Set Angle, θ to 30°. Complement, 90° − θ, returns 60° exactly — 1.0471975511965976 radians if you switch the unit selector, since 90° − 30° = 60° and 60° is π/3. Below it, sin(θ) = cos(complement) reports 0.49999999999999994: sine of 30° and cosine of 60° both round to the textbook 0.5, off by one part in ten quadrillion only because the sheet's engine runs in double-precision floating point rather than exact fractions.
The result checks itself two ways. Cosine of 60° computed on its own also lands on 0.5, so sin(30°) and cos(60°) arrive at the same figure by two independent routes around the same right triangle. Push θ up to 60° instead and the sheet reports the mirror image — Complement drops to 30°, and sin(60°) ≈ 0.8660 now lands on cos(30°), the pair swapping roles without the rule itself changing at all.
Questions
What is a cofunction identity?
It is an equation that trades a trig function of an angle for a different trig function of that angle's complement: sin θ = cos(90° − θ), cos θ = sin(90° − θ), and the matching pairs tan/cot and sec/csc. All six come from one right-triangle swap — opposite and adjacent trade places when you switch which acute angle you are measuring from.
Why is cosine called co-sine?
Because the co- prefix means 'of the complement'. Sine was already established when the English mathematician Edmund Gunter coined 'cosine' and 'cotangent' in the 1620s, naming each new function after its relationship to sine or tangent across a 90° split. The cofunction identity is not a side note attached to the word — it is the reason the word exists.
What happens to the identity at θ = 45°?
The angle becomes its own complement, since 45° + 45° = 90°, so sin(45°) and cos(45°) are forced to the same value, roughly 0.7071. It is the one self-paired point on the 0°-90° range this calculator covers, and the same symmetry behind an isosceles 45-45-90 triangle.
How is this different from a double-angle identity?
A double-angle identity rewrites a function of 2θ using θ alone, such as sin(2θ) = 2 sin θ cos θ — one angle, doubled. A cofunction identity instead relates two separate angles, θ and its complement 90° − θ, through two separate functions. They answer different questions, and neither one substitutes for the other.
Does the identity hold in radians as well as degrees?
Yes — swap 90° for π/2 and nothing else changes: sin θ = cos(π/2 − θ). The Angle field on this sheet accepts degrees, radians, or turns through its unit selector, and Complement always returns 90° − θ converted correctly into whichever unit you are reading.
What is the most common mistake with cofunction identities?
Reaching for the complement, 90° − θ, when a problem actually calls for the supplement, 180° − θ, which belongs to a different identity, sin θ = sin(180° − θ). The two rules look alike but come from different triangles: complements split a right angle, supplements split a straight one.