Pythagorean Identities

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

Advanced25 minLesson

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^2\theta + \cos^2\theta = 1

Identity 2:

1 + \tan^2\theta = \sec^2\theta

Identity 3:

1 + \cot^2\theta = \csc^2\theta

All three identities are true for any angle

\theta

where the functions are defined.

Try it now

What is the fundamental Pythagorean identity?

Worked Examples

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

Start with a point on the unit circle

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

Apply the unit circle equation

The unit circle is defined by $x^2 + y^2 = 1$

Substitute the coordinates

Replace $x$ with $\cos\theta$ and $y$ with $\sin\theta$ → $(\cos\theta)^2 + (\sin\theta)^2 = 1$

Write in standard form

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

Answer: The identity $\sin^2\theta + \cos^2\theta = 1$ is proven by the definition of the unit circle!

Common Mistakes

Forgetting to consider the quadrant when taking square roots

Why it's wrong: The equation $\cos^2\theta = \frac{16}{25}$ gives $\cos\theta = \pm\frac{4}{5}$ . 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 $\sin^2\theta$ as $\sin\theta^2$

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

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

Applying an identity where a function is undefined

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

sin(θ)

cos(θ)

tan(θ)

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.

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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 $A\sin\theta$ and $A\cos\theta$ , the total power is $A^2(\sin^2\theta + \cos^2\theta) = A^2$ , which is constant.

1Try It Yourself

A signal has components $3\sin\theta$ and $4\cos\theta$ .

What is the total amplitude?

Step 1: Write the mathematical expression

Use $(3)^2 + (4)^2$ 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 $x = A\cos(\omega t)$ and velocity is $v = -A\omega\sin(\omega t)$ , then $\frac{x^2}{A^2} + \frac{v^2}{A^2\omega^2} = \cos^2(\omega t) + \sin^2(\omega t) = 1$ .

2Try It Yourself

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

If the maximum position is $A$ , what fraction of $A\omega$ is the velocity?

Step 1: Write the mathematical expression

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

Key Takeaways

1The fundamental Pythagorean identity is $\sin^2\theta + \cos^2\theta = 1$
2Dividing by $\cos^2\theta$ gives $1 + \tan^2\theta = \sec^2\theta$
3Dividing by $\sin^2\theta$ gives $1 + \cot^2\theta = \csc^2\theta$
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

a^2 + b^2 = c^2

. On the unit circle, the legs are

\cos\theta

and

\sin\theta

, and the hypotenuse is 1.

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 $(\cos\theta, \sin\theta)$
Secant: $\sec\theta = \frac{1}{\cos\theta}$ , the reciprocal of cosine
Cosecant: $\csc\theta = \frac{1}{\sin\theta}$ , the reciprocal of sine
Cotangent: $\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{1}{\tan\theta}$ , the reciprocal of tangent

Formula Card

Fundamental Identity

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

The sum of sine squared and cosine squared always equals 1

Tangent-Secant Identity

1 + \tan^2\theta = \sec^2\theta

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

Cotangent-Cosecant Identity

1 + \cot^2\theta = \csc^2\theta

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