Introduction to Trigonometric Identities

Importance of Trigonometric Identities

• Essential for simplifying complex trigonometric problems.
• Helpful in higher-level calculus courses.
• Common misconception: perceived as difficult but actually simplify future calculations.

Basic Trigonometric Identities

• Reciprocal Identities:
  • ( \tan \theta = \frac{\sin \theta}{\cos \theta} )
  • ( \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta} )
  • ( \csc \theta = \frac{1}{\sin \theta} )
  • ( \sec \theta = \frac{1}{\cos \theta} )
• Pythagorean Identities:
  • ( \sin^2 \theta + \cos^2 \theta = 1 )
  • ( \tan^2 \theta + 1 = \sec^2 \theta )
  • ( \cot^2 \theta + 1 = \csc^2 \theta )
  • Variations:
    • ( \sin^2 \theta = 1 - \cos^2 \theta )
    • ( \cos^2 \theta = 1 - \sin^2 \theta )
    • ( \sec^2 \theta - 1 = \tan^2 \theta )
    • ( \sec^2 \theta - \tan^2 \theta = 1 )
• Even and Odd Identities:
  • Even: ( \cos(-\theta) = \cos(\theta) ), ( \sec(-\theta) = \sec(\theta) )
  • Odd: ( \sin(-\theta) = -\sin(\theta) ), ( \csc(-\theta) = -\csc(\theta) ), ( \tan(-\theta) = -\tan(\theta) ), ( \cot(-\theta) = -\cot(\theta) )

Simplification Process

1. Identify the Harder Side: Start simplification with the more complex side to make it manageable.
2. Combine Fractions: If more than one fraction, combine into a single fraction using a common denominator.
3. Convert to Sines and Cosines: Rewrite all expressions in terms of sine and cosine for simplification.
4. Use Known Identities: Utilize known identities to simplify expressions further.

Example Problems

• Example 1:

  • Expression: ( \tan \theta \cdot \csc \theta )
  • Rewrite using identities: ( \frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\sin \theta} )
  • Simplify: Result is ( \sec \theta )

• Example 2:

  • Expression: ( \frac{\sin \theta + \cos \theta}{\cos \theta} + \frac{\cos \theta - \sin \theta}{\sin \theta} )
  • Combine fractions: Use common denominator ( \sin \theta \cos \theta )
  • Simplify: Result is ( \csc \theta \cdot \sec \theta )

• Example 3:

  • Expression: ( \frac{\cos^2 \theta - 1}{\cos^2 \theta - \cos \theta} )
  • Factor numerator and denominator:
    • Numerator: ( (\cos \theta - 1)(\cos \theta + 1) )
    • Denominator: ( \cos \theta (\cos \theta - 1) )
  • Simplify: Result is ( 1 + \sec \theta )

Key Tips

• Use identities as two-way streets (both directions can simplify expressions).
• Factoring can help in simplifying expressions, particularly when terms are in sines/cosines.
• Simplification stops when no further identities apply.

Conclusion

• Understanding and applying these identities and simplification techniques is crucial for solving more complex trigonometry problems.
• Advanced topics will cover proving identities with detailed examples.