Lesson 5.3

The Primary Pythagorean Identity

The Pythagorean Theorem meets the unit circle to produce the single most important identity in trigonometry: .

Introduction

You used the Pythagorean Theorem in Chapter 1 to find missing sides of right triangles. On the unit circle, where the hypotenuse is always , that same theorem becomes an elegant relationship between sine and cosine that holds at every angle.

Past Knowledge

The Pythagorean Theorem: , and the unit circle where , .

Today's Goal

Prove and apply to find missing trig values.

Future Success

Two more Pythagorean identities (Lesson 5.4) are derived directly from this one.

Key Concepts

The Proof

Every point on the unit circle satisfies the equation . Since and , we substitute:

Rewritten in the conventional order:

Visualizing on the Unit Circle

For any angle , the point lies on the circle. The horizontal leg is , the vertical leg is , and the hypotenuse (radius) is .

Rearranged Forms

The identity can be solved for either function:

Solving for…	Result

When to Use This Identity

Use the Pythagorean identity whenever you know one trig function and need to find another, or when you see (or a rearranged form) in an expression you're simplifying.

Worked Examples

Basic

Finding Cosine Given Sine

Question: If and is in Quadrant I, find .

Step 1: Substitute into the Pythagorean identity:

Step 2: Simplify:

Step 3: Take the square root. In QI, cosine is positive:

Final Answer:

Intermediate

Using the Rearranged Form to Simplify

Question: Simplify .

Step 1: Recognize the rearranged Pythagorean identity:

Final Answer:

Advanced

Finding All Six Trig Values

Question: Given and is in Quadrant III, find all six trig values.

Step 1: Find using the Pythagorean identity:

In QIII, sine is negative: .

Step 2: Use quotient and reciprocal identities:

Final Answer: , , , , ,

Common Pitfalls

Forgetting the ± Sign

When you solve , the result is . You must use the quadrant to determine the correct sign. Skipping this step gives incorrect answers half the time.

Writing

The identity involves squares: . The non-squared version is not an identity (try ).

Real-Life Applications

Navigation & GPS

GPS satellites compute your position using trigonometric relationships on the surface of the Earth (modeled as a sphere). The Pythagorean identity ensures that latitude and longitude calculations remain consistent — since every point on a unit sphere satisfies , the system can reliably convert between different coordinate frames without accumulating error.