Why are they called cofunction identities?

The prefix co- in cosine, cotangent, and cosecant stands for complementary. Cosine is the sine of the complementary angle. Similarly, cotangent is the tangent of the complementary angle, and cosecant is the secant of the complementary angle.

How do cofunction identities work on the unit circle?

On the unit circle, the point at angle A is (cosA, sinA). The point at 90-A is (sinA, cosA). The x and y coordinates swap, which is why cos(90-A) = sinA and sin(90-A) = cosA. This reflects the point across the line y=x.

Where are cofunction identities used?

They are used extensively in calculus (integration by substitution), physics (analyzing complementary angles in force diagrams), engineering (signal phase shifting), and simplifying trigonometric expressions in proofs.

IP331.com | Online Tools

Algebra Tools Geometry Tools Trigonometry Tools Matrix Tools Function Tools Number Theory Tools Base Converter Electronics & Physics Statistics Tools Life & Finance

Home›Trigonometry Tools›Cofunction Identity Solver

Cofunction Identity Solver

Enter an angle to apply cofunction identities: sin -> cos, tan -> cot, sec -> csc

Angle A (degrees)

Cofunction Identities

sin(90 - A) = cosA

cos(90 - A) = sinA

tan(90 - A) = cotA

cot(90 - A) = tanA

sec(90 - A) = cscA

csc(90 - A) = secA

Cofunction identities show that the sine of an angle equals the cosine of its complement, and vice versa. The name cofunction comes from the complement relationship: cosine = sine of the complement.

⚠ Cofunction identities work only for complementary angles (sum = 90 degrees). For angles outside 0-90, use reference angles to find the equivalent acute cofunction.

What Are Cofunction Identities?

Cofunction identities describe the relationship between trigonometric functions at complementary angles. The term cofunction comes from the fact that the cosine is the sine of the complementary angle.

Sine and Cosine

sin(90-A) = cosA. If A=30, sin60 = cos30 = 0.866. The names reveal the relationship: co-sine = complement sine.

Tangent and Cotangent

tan(90-A) = cotA. If A=45, tan45 = cot45 = 1. Co-tangent = complement tangent.

Unit Circle View

Reflecting a point across y=x swaps the (x,y) coordinates, which is exactly what cofunctions do: (cosA, sinA) becomes (sinA, cosA).

Derivation

From the subtraction formula: cos(90-A) = cos90cosA + sin90sinA = 0xcosA + 1xsinA = sinA.

Teaching Example: A = 30 degrees. Complementary = 60 degrees. sin(60) = cos(30) = 0.866. cos(60) = sin(30) = 0.5. tan(60) = cot(30) = 1.732. All verified: sin60 = 0.866, cos30 = 0.866. cos60 = 0.5, sin30 = 0.5.

Applications

Trig Simplification Calculus Integration Physics Forces Engineering Signal Phase Shift Proof Writing

FAQs about Cofunction Identities

What are cofunction identities?▼

sin(90-A)=cosA, cos(90-A)=sinA, tan(90-A)=cotA. They relate trig functions of complementary angles.

What is the cofunction of sin?▼

The cofunction of sine is cosine. sin(90-A) = cosA. The word cosine literally means complement sine.

Do cofunction identities work for radians?▼

Yes, in radians: sin(pi/2 - A) = cosA, cos(pi/2 - A) = sinA. The tool converts degrees to radians internally.

What is sec(90-A)?▼

sec(90-A) = cscA. Similarly, csc(90-A) = secA. This follows from the reciprocal relationship with sine and cosine.

More Trigonometry Tools

Cosine Calculator

Calculate cos(x) for angles in degrees or radians with unit circle explanation

›

Phase Shift Calculator for Trig Functions

Compute the horizontal translation and starting point of trigonometric graphs

›

Angle to Grade Calculator

Convert slope angle in degrees to percent grade using tangent

›

Trig Values Calculator

Calculate all six trigonometric function values for an angle in degrees

›

Secant Graph Calculator

Find vertical stretch, period, phase shift, and midline for a secant graph

›

Exact Value of Trig Ratio Calculator

Find exact sine, cosine, and tangent values for common angles using special triangles

›