Trigonometric Identities and Simplification Techniques

Identities

  • Definition of an Identity: An equation that is true for all values of the variable(s).

Fundamental Identities

  • Quotient Identities:
    • \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
    • \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
  • Reciprocal Identities:
    • \csc(\theta) = \frac{1}{\sin(\theta)}
    • \sec(\theta) = \frac{1}{\cos(\theta)}
    • \cot(\theta) = \frac{1}{\tan(\theta)}
  • Pythagorean Identities:
    • \sin^2(\theta) + \cos^2(\theta) = 1
    • \tan^2(\theta) + 1 = \sec^2(\theta)
    • \cot^2(\theta) + 1 = \csc^2(\theta)

Even-Odd Identities

  • Sine: \sin(-a) = -\sin(a)
  • Cosine: \cos(-a) = \cos(a)
  • Tangent: \tan(-a) = -\tan(a)

Simplifying Trigonometric Expressions

  • Purpose: To manipulate expressions to get a simpler or more useful form.
  • Techniques:
    1. Use Trig Identities: Replace expressions using known identities.
    2. Algebraic Manipulation: Use distributive law, factor, combine like terms, find common denominators.
    3. Various Techniques: Other algebraic manipulations as necessary.

Example Problems

  1. Example: \cos(x) \cdot \sec(x) + \tan(x)

    • Substitute: \sec(x) = \frac{1}{\cos(x)}
    • Result: 1 + \sin(x)

  2. Example: \sin(x) \tan(x) (\csc(x) + \cot(x))

    • Result: \tan(x) + \sin(x)

  3. Example: 1 - \sec^2(x)

    • Substitute \sec^2(x) = \tan^2(x) + 1
    • Result: -\tan^2(x)

Factor and Simplify Examples

  • Example: \sin^3(x) + \cos^2(x)\sin(x)

    • Factor: \sin(x)(\sin^2(x) + \cos^2(x)) = \sin(x)

  • Example: \sec^2(x) \cot(x) - \sec(x) \sin(x)

    • Factor: \sec(x)(\cot(x) - \sin(x))

Finding Common Denominators

  • Example: \frac{\sin(\theta)}{\cos(\theta)} + \frac{\cos(\theta)}{\sin(\theta)}
    • Result: \frac{\sin^2(\theta) + \cos^2(\theta)}{\sin(\theta) \cos(\theta)} = \frac{1}{\sin(\theta) \cos(\theta)}

Homework and Exam Information

  • Worksheet Due: Wednesday, April 23
  • Next Exam: Friday, May 2

Additional Notes

  • Always memorize and be familiar with fundamental identities as they are essential for proofs and simplifications.