Cofunction Identities

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

Advanced25 minLesson
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Definition

Cofunction identities relate trigonometric functions of complementary angles. Two angles are complementary if they sum to 90°90° (or π2\frac{\pi}{2} radians).
The six cofunction identities are:
sin⁡(θ)=cos⁡(90°−θ)\sin(\theta) = \cos(90° - \theta)
cos⁡(θ)=sin⁡(90°−θ)\cos(\theta) = \sin(90° - \theta)
tan⁡(θ)=cot⁡(90°−θ)\tan(\theta) = \cot(90° - \theta)
cot⁡(θ)=tan⁡(90°−θ)\cot(\theta) = \tan(90° - \theta)
sec⁡(θ)=csc⁡(90°−θ)\sec(\theta) = \csc(90° - \theta)
csc⁡(θ)=sec⁡(90°−θ)\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⁡(θ)\sin(\theta)?

Worked Examples

Verify that sin⁡(30°)=cos⁡(60°)\sin(30°) = \cos(60°)
1

Calculate sin⁡(30°)\sin(30°)

From the unit circle or special triangles: sin⁡(30°)=12\sin(30°) = \frac{1}{2} → sin⁡(30°)=0.5\sin(30°) = 0.5
2

Find the complement of 30°30°

90°−30°=60°90° - 30° = 60° → Complement is 60°60°
3

Calculate cos⁡(60°)\cos(60°)

From the unit circle or special triangles: cos⁡(60°)=12\cos(60°) = \frac{1}{2} → cos⁡(60°)=0.5\cos(60°) = 0.5
4

Compare the values

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

Common Mistakes

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

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

Correct: Always remember: cofunction = complement = 90°90°. Use sin⁡(θ)=cos⁡(90°−θ)\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°90° and π2\frac{\pi}{2} are the same, but you must be consistent.

Correct: Stick to one unit system: either 90°−θ90° - \theta or π2−θ\frac{\pi}{2} - \theta.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Complementary and Supplementary Angle Identities | Examples

by Understand The Math

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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

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0

sin(θ)

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cos(θ)

1

tan(θ)

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Coordinates (cos, sin)

(1, 0)

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Practice Problems

15 problems
Problem 1 of 15
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What is the cofunction of sin⁡(θ)\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°)\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°35°, the angle of depression from the top is also 35°35°, and the line of sight makes 55°55° with the vertical wall, so sin⁡(35°)=cos⁡(55°)\sin(35°) = \cos(55°).

1Try It Yourself

A surveyor at point A measures an angle of elevation of 40°40° to the top of a building. The angle of depression from the building top to point A is the same 40°40°. What angle does the line of sight make with the vertical wall of the building?

What angle is there at the top of the building between the wall and the line of sight to point A?

Step 1: Write the mathematical expression

The angle of depression and the angle with the wall are complementary: 90°−40°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⁡(ωt)\cos(\omega t) can be written as sin⁡(ωt+90°)\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⁡(θ)\sin(\theta) and needs to express it as a cosine function.

How can sin⁡(θ)\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°90°)
  • 2The six pairs: sin⁡↔cos⁡\sin \leftrightarrow \cos, tan⁡↔cot⁡\tan \leftrightarrow \cot, sec⁡↔csc⁡\sec \leftrightarrow \csc
  • 3Key formula: sin⁡(θ)=cos⁡(90°−θ)\sin(\theta) = \cos(90° - \theta) and vice versa
  • 4The "co-" prefix indicates the cofunction: cosine is the cofunction of sine
  • 5In radians: replace 90°90° with π2\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.'
The word comes from 'complementary function.' The cosine is the sine of the complementary angle. The prefix 'co-' means 'complement of.'
Yes! While they're easiest to visualize with acute angles in a right triangle, the identities hold for all angles. For example, sin⁡(120°)=cos⁡(−30°)\sin(120°) = \cos(-30°) because 120°+(−30°)=90°120° + (-30°) = 90°.
Look for the 'co-' prefix: sine pairs with COsine, tangent pairs with COtangent, secant pairs with COsecant. The function without 'co-' pairs with the one that has it.

Glossary

Cofunction
A trigonometric function whose value equals another function of the complementary angle
Complementary angles
Two angles that sum to 90°90° (or π2\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⁡(θ)=cos⁡(90°−θ)\sin(\theta) = \cos(90° - \theta)

Sine equals cosine of complement

Cosine-Sine

cos⁡(θ)=sin⁡(90°−θ)\cos(\theta) = \sin(90° - \theta)

Cosine equals sine of complement

Tangent-Cotangent

tan⁡(θ)=cot⁡(90°−θ)\tan(\theta) = \cot(90° - \theta)

Tangent equals cotangent of complement

Cotangent-Tangent

cot⁡(θ)=tan⁡(90°−θ)\cot(\theta) = \tan(90° - \theta)

Cotangent equals tangent of complement

Secant-Cosecant

sec⁡(θ)=csc⁡(90°−θ)\sec(\theta) = \csc(90° - \theta)

Secant equals cosecant of complement

Cosecant-Secant

csc⁡(θ)=sec⁡(90°−θ)\csc(\theta) = \sec(90° - \theta)

Cosecant equals secant of complement

Today's challengeOne problem, three tries

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