Explanation

The unit circle is a circle with:

• Radius = 1

• Center at (0, 0)

For any angle θ:

• cos(θ) is the x-coordinate

• sin(θ) is the y-coordinate

A trigonometric identity is an equation that is true for all θ where it is defined.

The most important identity:
sin²(θ) + cos²(θ) = 1

This comes directly from the Pythagorean Theorem:
x² + y² = 1

Other common identities:

• tan(θ) = sin(θ) / cos(θ)

• 1 + tan²(θ) = sec²(θ)

• 1 + cot²(θ) = csc²(θ)

Quiz

1. What does cos(θ) represent on the unit circle?

2. State the Pythagorean identity.

3. How is tan(θ) defined using sine and cosine?

4. Is sin²(θ) + cos²(θ) = 1 sometimes or always true?

5. Which identity comes from the Pythagorean Theorem?

Answer Key

1. x-coordinate

2. sin²(θ) + cos²(θ) = 1

3. sin(θ) / cos(θ)

4. Always

5. sin²(θ) + cos²(θ) = 1