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⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
    • cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
  • Reciprocal Identities:
    • csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}
    • sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}
    • cot⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{1}{\tan(\theta)}
  • Pythagorean Identities:
    • sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1
    • tan⁡2(θ)+1=sec⁡2(θ)\tan^2(\theta) + 1 = \sec^2(\theta)
    • cot⁡2(θ)+1=csc⁡2(θ)\cot^2(\theta) + 1 = \csc^2(\theta)
Even-Odd Identities
  • Sine: sin⁡(−a)=−sin⁡(a)\sin(-a) = -\sin(a)
  • Cosine: cos⁡(−a)=cos⁡(a)\cos(-a) = \cos(a)
  • Tangent: tan⁡(−a)=−tan⁡(a)\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)⋅sec⁡(x)+tan⁡(x)\cos(x) \cdot \sec(x) + \tan(x)

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

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

    • Result: tan⁡(x)+sin⁡(x)\tan(x) + \sin(x)

  3. Example: 1−sec⁡2(x)1 - \sec^2(x)

    • Substitute sec⁡2(x)=tan⁡2(x)+1\sec^2(x) = \tan^2(x) + 1
    • Result: −tan⁡2(x)-\tan^2(x)
Factor and Simplify Examples

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

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

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

    • Factor: sec⁡(x)(cot⁡(x)−sin⁡(x))\sec(x)(\cot(x) - \sin(x))
Finding Common Denominators
  • Example: sin⁡(θ)cos⁡(θ)+cos⁡(θ)sin⁡(θ)\frac{\sin(\theta)}{\cos(\theta)} + \frac{\cos(\theta)}{\sin(\theta)}
    • Result: sin⁡2(θ)+cos⁡2(θ)sin⁡(θ)cos⁡(θ)=1sin⁡(θ)cos⁡(θ)\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.