Pythagorean Identities

Learn the three fundamental Pythagorean identities and how to derive and apply them.

Advanced25 minLesson
Printable worksheet

Definition

The Pythagorean identities are three fundamental equations in trigonometry that relate the squares of trigonometric functions. They are derived from the Pythagorean theorem.

The Three Pythagorean Identities

Identity 1 (Fundamental):
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
Identity 2:
1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta
Identity 3:
1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta
All three identities are true for any angle θ\theta where the functions are defined.

Try it now

What is the fundamental Pythagorean identity?

Worked Examples

Prove that sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 using the unit circle.
1

Start with a point on the unit circle

Any point on the unit circle has coordinates (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) → (x,y)=(cos⁡θ,sin⁡θ)(x, y) = (\cos\theta, \sin\theta)
2

Apply the unit circle equation

The unit circle is defined by x2+y2=1x^2 + y^2 = 1
3

Substitute the coordinates

Replace xx with cos⁡θ\cos\theta and yy with sin⁡θ\sin\theta → (cos⁡θ)2+(sin⁡θ)2=1(\cos\theta)^2 + (\sin\theta)^2 = 1
4

Write in standard form

Use exponent notation: sin⁡2θ+cos⁡2θ\sin^2\theta + \cos^2\theta → sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 ✓

Common Mistakes

Forgetting to consider the quadrant when taking square roots

Why it's wrong: The equation cos⁡2θ=1625\cos^2\theta = \frac{16}{25} gives cos⁡θ=±45\cos\theta = \pm\frac{4}{5}. Students often forget to determine which sign applies.

Correct: Always check which quadrant the angle is in to determine the sign of the trig function.

Writing sin⁡2θ\sin^2\theta as sin⁡θ2\sin\theta^2

Why it's wrong: sin⁡2θ\sin^2\theta means (sin⁡θ)2(\sin\theta)^2, not sin⁡(θ2)\sin(\theta^2). The notation is shorthand for squaring the result of the sine function.

Correct: sin⁡2θ=(sin⁡θ)2\sin^2\theta = (\sin\theta)^2, meaning "find sine of theta, then square the result."

Applying an identity where a function is undefined

Why it's wrong: The identity 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta is undefined when cos⁡θ=0\cos\theta = 0 (at θ=90°,270°\theta = 90°, 270°, etc.).

Correct: Check that the functions in the identity are defined for the given angle.

Watch a video explanation

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

Pythagorean Identities Explained: Derivation and Examples

by Understand The Math

Watch on YouTube

Interactive Visual

Unit Circle

Degrees

0°

Radians

0

sin(θ)

0

cos(θ)

1

tan(θ)

0

Coordinates (cos, sin)

(1, 0)

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

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

Remember: SOH-CAH-TOA

SOH

Sin = Opp / Hyp

CAH

Cos = Adj / Hyp

TOA

Tan = Opp / Adj

Click on sin, cos, or tan to highlight the relevant sides of the triangle.

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

15 problems
Problem 1 of 15
Easy
What is the fundamental Pythagorean identity?

Why It Matters

Pythagorean identities are essential tools in trigonometry:
  • Simplifying expressions: Convert between trig functions to simplify complex expressions
  • Solving equations: Transform trigonometric equations into solvable forms
  • Calculus: Used extensively in integration and differentiation
  • Physics: Appear in wave equations, oscillations, and electromagnetic theory
  • Verifying identities: Serve as building blocks for proving other trig identities
Without these identities, many advanced mathematical techniques would be impossible!

Real World Applications

Signal Processing

Engineers use Pythagorean identities when analyzing radio and audio signals that are modeled by sine and cosine waves.

Example:

When combining two signals Asin⁡θA\sin\theta and Acos⁡θA\cos\theta, the total power is A2(sin⁡2θ+cos⁡2θ)=A2A^2(\sin^2\theta + \cos^2\theta) = A^2, which is constant.

1Try It Yourself

A signal has components 3sin⁡θ3\sin\theta and 4cos⁡θ4\cos\theta.

What is the total amplitude?

Step 1: Write the mathematical expression

Use (3)2+(4)2(3)^2 + (4)^2 to find the amplitude squared:

Physics: Simple Harmonic Motion

The position and velocity of an oscillating object are related through sine and cosine. Their energy relationship uses the Pythagorean identity.

Example:

If position is x=Acos⁡(ωt)x = A\cos(\omega t) and velocity is v=−Aωsin⁡(ωt)v = -A\omega\sin(\omega t), then x2A2+v2A2ω2=cos⁡2(ωt)+sin⁡2(ωt)=1\frac{x^2}{A^2} + \frac{v^2}{A^2\omega^2} = \cos^2(\omega t) + \sin^2(\omega t) = 1.

2Try It Yourself

A pendulum has position x=0.6Ax = 0.6A at some instant.

If the maximum position is AA, what fraction of AωA\omega is the velocity?

Step 1: Write the mathematical expression

Use sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 with cos⁡θ=0.6\cos\theta = 0.6:

Key Takeaways

  • 1The fundamental Pythagorean identity is sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
  • 2Dividing by cos⁡2θ\cos^2\theta gives 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta
  • 3Dividing by sin⁡2θ\sin^2\theta gives 1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta
  • 4These identities let you convert between trig functions and simplify expressions
  • 5Always consider the quadrant when taking square roots

Frequently Asked Questions

They derive from the Pythagorean theorem a2+b2=c2a^2 + b^2 = c^2. On the unit circle, the legs are cos⁡θ\cos\theta and sin⁡θ\sin\theta, and the hypotenuse is 1.
They derive from the Pythagorean theorem a2+b2=c2a^2 + b^2 = c^2. On the unit circle, the legs are cos⁡θ\cos\theta and sin⁡θ\sin\theta, and the hypotenuse is 1.
The fundamental identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 works for all angles. The other two identities work except where tan, cot, sec, or csc are undefined (division by zero).
Memorize only sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1. Derive the others by dividing by cos⁡2θ\cos^2\theta or sin⁡2θ\sin^2\theta as needed.

Glossary

Pythagorean identity
A trigonometric equation derived from the Pythagorean theorem, relating squares of trig functions
Unit circle
A circle with radius 1 centered at the origin, where any point is (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta)
Secant
sec⁡θ=1cos⁡θ\sec\theta = \frac{1}{\cos\theta}, the reciprocal of cosine
Cosecant
csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta}, the reciprocal of sine
Cotangent
cot⁡θ=cos⁡θsin⁡θ=1tan⁡θ\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{1}{\tan\theta}, the reciprocal of tangent

Formula Card

Fundamental Identity

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

The sum of sine squared and cosine squared always equals 1

Tangent-Secant Identity

1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta

Derived by dividing the fundamental identity by $\cos^2\theta$

Cotangent-Cosecant Identity

1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta

Derived by dividing the fundamental identity by $\sin^2\theta$

Today's challengeOne problem, three tries

More in This Topic

Reciprocal Identities

Advanced25 min

Even and Odd Trigonometric Identities

Advanced25 min

Cofunction Identities

Advanced25 min

Double Angle Identities

Advanced30 min