Periodic Functions: Background You’ll Need 3

  • Identify values where trigonometric functions are undefined
Trigonometric functions are undefined when their denominators are zero or when they involve a ratio that cannot exist.

Examples of when trigonometric functions are undefined:

  • [latex]\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}[/latex] → undefined when [latex]\cos(\theta) = 0[/latex]

  • [latex]\sec(\theta) = \frac{1}{\cos(\theta)}[/latex] → undefined when [latex]\cos(\theta) = 0[/latex]

  • [latex]\csc(\theta) = \frac{1}{\sin(\theta)}[/latex] → undefined when [latex]\sin(\theta) = 0[/latex]

  • [latex]\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}[/latex] → undefined when [latex]\sin(\theta) = 0[/latex]

sine and cosine have infinite domains and will never be undefined, but secant, cosecant, tangent, and cotangent are all undefined for some angles in their domain.