Algebra 101 - Unit 16

Introduction

Trigonometric identities are equations involving trigonometric functions that are true for all valid angles. Mastery of these identities allows simplification of expressions, solving equations, and proving other identities.

Fundamental Identities

• Pythagorean Identities:
  • sin² θ + cos² θ = 1
  • 1 + tan² θ = sec² θ
  • 1 + cot² θ = csc² θ
• Reciprocal Identities:
  • sin θ = 1/csc θ
  • cos θ = 1/sec θ
  • tan θ = 1/cot θ
  • csc θ = 1/sin θ
  • sec θ = 1/cos θ
  • cot θ = 1/tan θ
• Quotient Identities:
  • tan θ = sin θ / cos θ
  • cot θ = cos θ / sin θ

Co-Function Identities

• sin(90° − θ) = cos θ
• cos(90° − θ) = sin θ
• tan(90° − θ) = cot θ
• cot(90° − θ) = tan θ
• sec(90° − θ) = csc θ
• csc(90° − θ) = sec θ

Strategy for Using Identities

1. Identify which identity or combination of identities may simplify the expression or equation.
2. Rewrite all functions in terms of sine and cosine when possible.
3. Factor, combine like terms, or apply Pythagorean identities to simplify.
4. Check the domain to ensure the identity is valid for all angles considered.

Examples

Example 1 — Simplify:

Simplify (1 − cos² θ)/sin θ → sin² θ / sin θ → sin θ

Example 2 — Verify Identity:

Verify tan² θ + 1 = sec² θ → Using Pythagorean identity: 1 + tan² θ = sec² θ ✔

Example 3 — Rewrite Expression:

Simplify csc θ − sin θ / sin θ → (1/sin θ − sin θ)/sin θ → (1 − sin² θ)/sin² θ → cos² θ / sin² θ → cot² θ

Practice Problems

1. Simplify (1 − cos² θ)/sin θ.
2. Verify the identity: 1 + cot² θ = csc² θ.
3. Simplify tan θ · cot θ.
4. Rewrite sec² θ − 1 in terms of tan θ.
5. Prove that sin θ / (1 + cos θ) = (1 − cos θ) / sin θ.