Cofunction Identities

Learn how sine and cosine, tangent and cotangent, secant and cosecant are related through complementary angles.

Advanced25 minLesson

Definition

Cofunction identities relate trigonometric functions of complementary angles. Two angles are complementary if they sum to

90°

(or

\frac{\pi}{2}

radians).

The six cofunction identities are:

\sin(\theta) = \cos(90° - \theta)

\cos(\theta) = \sin(90° - \theta)

\tan(\theta) = \cot(90° - \theta)

\cot(\theta) = \tan(90° - \theta)

\sec(\theta) = \csc(90° - \theta)

\csc(\theta) = \sec(90° - \theta)

The word "cofunction" comes from "complementary function" - the cosine is the sine of the complement!

Try it now

What is the cofunction of

\sin(\theta)

Worked Examples

Verify that $\sin(30°) = \cos(60°)$

Calculate $\sin(30°)$

From the unit circle or special triangles: $\sin(30°) = \frac{1}{2}$ → $\sin(30°) = 0.5$

Find the complement of $30°$

$90° - 30° = 60°$ → Complement is $60°$

Calculate $\cos(60°)$

From the unit circle or special triangles: $\cos(60°) = \frac{1}{2}$ → $\cos(60°) = 0.5$

Compare the values

$\sin(30°) = 0.5 = \cos(60°)$ → Identity verified!

Answer: $\sin(30°) = \cos(60°) = \frac{1}{2}$ is verified

Common Mistakes

Forgetting that complementary angles sum to $90°$ , not $180°$

Why it's wrong: Supplementary angles sum to $180°$ , but cofunction identities specifically use complementary angles ( $90°$ ).

Correct: Always remember: cofunction = complement = $90°$ . Use $\sin(\theta) = \cos(90° - \theta)$ .

Confusing which functions are cofunctions of each other

Why it's wrong: It's easy to mix up pairs. The prefix "co-" is the key: sine/cosine, tangent/cotangent, secant/cosecant.

Correct: Look for the "co-" prefix: sin ↔ cosin(e), tan ↔ cotan(gent), sec ↔ cosec(ant).

Using degrees in one function and radians in another

Why it's wrong: Mixing units leads to incorrect results. $90°$ and $\frac{\pi}{2}$ are the same, but you must be consistent.

Correct: Stick to one unit system: either $90° - \theta$ or $\frac{\pi}{2} - \theta$ .

Interactive Visual

Right Triangle Trigonometry

θ =30°

5°45°85°

sin(θ)

0.5

Opp / Hyp

cos(θ)

√3/2

Adj / Hyp

tan(θ)

√3/3

Opp / Adj

Move the slider to change the angle and see how trigonometric ratios change.

Unit Circle

Degrees

0°

Radians

sin(θ)

cos(θ)

tan(θ)

Coordinates (cos, sin)

(1, 0)

Click and drag to rotate the angle around the unit circle.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems

Problem 1 of 15

Easy

What is the cofunction of $\sin(\theta)$ ?

Why It Matters

Cofunction identities are powerful tools in trigonometry:

Simplifying expressions: Convert between functions to combine or simplify terms
Solving equations: Find equivalent forms that are easier to solve
Right triangle geometry: Understand why the acute angles in a right triangle have related trig values
Calculus preparation: These identities are essential for integration and differentiation
Engineering applications: Used in signal processing, physics, and structural analysis

Once you see that

\sin(30°) = \cos(60°)

, you'll understand the deep symmetry in trigonometry!

Real World Applications

Right Triangle Surveying

Surveyors use cofunction identities when measuring angles from different reference points.

Example:

If a surveyor measures an angle of elevation of $35°$ , the complementary angle of depression from the top is $55°$ , and $\sin(35°) = \cos(55°)$ .

1Try It Yourself

A surveyor at point A measures an angle of elevation of $40°$ to the top of a building. What is the angle of depression from the building top to point A?

What angle would someone at the top measure looking down to point A?

Step 1: Write the mathematical expression

The angles are complementary: $90° - 40°$

Signal Processing

In electronics, sine and cosine waves are used to represent signals. Cofunction identities help convert between them.

Example:

A cosine signal $\cos(\omega t)$ can be written as $\sin(\omega t + 90°)$ , representing the same wave shifted by a quarter period.

2Try It Yourself

An audio engineer has a sine wave signal $\sin(\theta)$ and needs to express it as a cosine function.

How can $\sin(\theta)$ be written using cosine?

Step 1: Write the mathematical expression

Use the cofunction identity...

Key Takeaways

1Cofunction identities relate trig functions of complementary angles (angles that sum to $90°$ )
2The six pairs: $\sin \leftrightarrow \cos$ , $\tan \leftrightarrow \cot$ , $\sec \leftrightarrow \csc$
3Key formula: $\sin(\theta) = \cos(90° - \theta)$ and vice versa
4The "co-" prefix indicates the cofunction: cosine is the cofunction of sine
5In radians: replace $90°$ with $\frac{\pi}{2}$

Frequently Asked Questions

The word comes from 'complementary function.' The cosine is the sine of the complementary angle. The prefix 'co-' means 'complement of.'

Glossary

Cofunction: A trigonometric function whose value equals another function of the complementary angle
Complementary angles: Two angles that sum to $90°$ (or $\frac{\pi}{2}$ radians)
Identity: An equation that is true for all values of the variable
Cofunction pairs: sin/cos, tan/cot, sec/csc - each pair relates through complementary angles

Formula Card

Sine-Cosine

\sin(\theta) = \cos(90° - \theta)

Sine equals cosine of complement

Cosine-Sine

\cos(\theta) = \sin(90° - \theta)

Cosine equals sine of complement

Tangent-Cotangent

\tan(\theta) = \cot(90° - \theta)

Tangent equals cotangent of complement

Cotangent-Tangent

\cot(\theta) = \tan(90° - \theta)

Cotangent equals tangent of complement

Secant-Cosecant

\sec(\theta) = \csc(90° - \theta)

Secant equals cosecant of complement

Cosecant-Secant

\csc(\theta) = \sec(90° - \theta)

Cosecant equals secant of complement