Overview

This topic explores key trigonometric identities and how they are used to simplify expressions and solve equations. It also introduces real-world applications such as angle of elevation/depression, harmonic motion, and wave modeling.

Key Identities and Applications

• Pythagorean Identities:
  • sin^2(θ) + cos^2(θ) = 1
  • 1 + tan^2(θ) = sec^2(θ)
  • 1 + cot^2(θ) = csc^2(θ)
• Reciprocal Identities:
  • 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: Used to relate complementary angles (e.g., sin(θ) = cos(90° - θ))
• Applications:
  • Solving right and oblique triangles
  • Modeling periodic phenomena
  • Calculating heights and distances with angles of elevation and depression

Step-by-Step Example

Problem: Verify the identity: 1 + tan^2(θ) = sec^2(θ)

Step 1: Start with the left-hand side:
1 + tan^2(θ) = 1 + (sin^2(θ)/cos^2(θ))

Step 2: Combine using a common denominator:
1 + tan^2(θ) = (cos^2(θ) + sin^2(θ))/cos^2(θ)

Step 3: Apply the Pythagorean identity:
cos^2(θ) + sin^2(θ) = 1, so:
1 + tan^2(θ) = 1/cos^2(θ) = sec^2(θ)

Answer: Verified

Quick Tip