Cofunction Identities

Cofunction identities show the relationship between the different trigonometric functions and their complementary angles. In this guide, you will learn more about cofunction identities.

Tutor-style math help

Cofunction Identities: what to notice and how to work it

Trigonometry skill

Trigonometry connects an angle to a triangle ratio, a unit-circle coordinate, or a repeating graph. Choosing the right picture makes the problem much easier.

What to notice first

Decide whether the problem is triangle-based, circle-based, or graph-based. Then use the matching definition.

Common student mistake

Do not mix degrees and radians. The angle unit must match the formula, graph scale, or calculator setting.

Key formulas and cues

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)

\(\tan\theta=\frac{\sin\theta}{\cos\theta}\)

\(\sin^2\theta+\cos^2\theta=1\)

A reliable path

Choose the modelUse a right triangle, the unit circle, or a transformed graph.
Track unitsConvert degrees and radians when needed.
Use identitiesReplace complicated trig expressions with equivalent simpler ones.

Worked examples

Right-triangle sine

Example: opposite = 5, hypotenuse = 13

Sine is opposite over hypotenuse.
Substitute 5 and 13.
Leave the ratio simplified.

Answer: \(\sin\theta=\frac5{13}\)

Unit-circle cosine

Example: \(\cos(0)\)

At angle 0, the point is (1, 0).
Cosine is the x-coordinate.
Read the x-value.

Answer: \(1\)

Open pop-up practice

Try one before moving on

Try: In a right triangle, tangent equals which ratio?

Answer: Opposite over adjacent.

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

A step-by-step guide to cofunction identities

Cofunction identities are trigonometric identities that show a relationship between trigonometric functions and complementary angles.

We have six identities that can be obtained using right triangles, the angle sum property of a triangle, and trigonometric ratio formulas.

The cofunction identities establish a relationship between trigonometric functions \(sin\) and \(cos\), \(tan\) and \(cot\), and \(sec\) and \(csc\). These functions are known as cofunctions of each other.

We can write cofunction identities in terms of radians and degrees because these are the units of angle measurement.