Trig Graphs and Early Identities Quiz Notes

Trig Graphs and Early Identities Quiz Outline

  • Date of Quiz: February 26, 2026
  • Course: Precalculus 12

1. Trig Graphs - csc, sec, & tan

  • Graphs of Trigonometric Functions:
    • Understand the basic properties and shapes of the cosecant (csc), secant (sec), and tangent (tan) functions.

Cosecant Function (csc)

  • Definition: The cosecant function is the reciprocal of the sine function, defined as:
    • csc(x) = \frac{1}{sin(x)}
  • Key features to note when sketching:
    • Asymptotes at points where (sin(x) = 0).
    • Intervals of increase and decrease.
    • Maximum and minimum values for the csc function.

Secant Function (sec)

  • Definition: The secant function is the reciprocal of the cosine function, defined as:
    • sec(x) = \frac{1}{cos(x)}
  • Key features to note:
    • Asymptotes at points where (cos(x) = 0).
    • Behavior in relation to the cosine function.

Tangent Function (tan)

  • Definition: The tangent function is the ratio of sine to cosine, defined as:
    • tan(x) = \frac{sin(x)}{cos(x)}
  • Key features to note:
    • Asymptotes at points where (cos(x) = 0).
    • Periodicity and repeating cycles of the tangent function.

2. Using Identities to Find Exact Trig Values

  • Importance of Trigonometric Identities:
    • Identities are fundamental tools in trigonometry that allow for simplification and solving of equations.
    • The quiz will provide situations where application of identities is necessary to find exact values.

Types of Identities Covered:

  • Compound Identities:

    • Useful for transforming compound angles into simpler forms.
    • Examples include:
    • sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
    • cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

  • Cofunction Identities:

    • Relate the trigonometric functions of complementary angles.
    • Examples include:
    • sin(90° - x) = cos(x)
    • tan(90° - x) = cot(x)

  • Double Angle Identities:

    • Used to express trigonometric functions of double angles in terms of single angles.
    • Examples include:
    • sin(2x) = 2sin(x)cos(x)
    • cos(2x) = cos^2(x) - sin^2(x)

  • Half Angle Identities:

    • Applied to express trigonometric functions for half of an angle.
    • Examples include:
    • sin\left(\frac{x}{2}\right) = \sqrt{\frac{1 - cos(x)}{2}}
    • cos\left(\frac{x}{2}\right) = \sqrt{\frac{1 + cos(x)}{2}}

Additional Preparation

  • Practice Tasks:
    • Complete the graphing practice questions to reinforce understanding of the shapes and characteristics of the trig functions.
    • Work on the identity homework from the "To Find Exact Values" handouts to enhance familiarity with applying identities, ensuring readiness for the quiz.
  • Clarifications:
    • The quiz will NOT require students to write the trigonometric equations from a graph; focus will be on sketching graphs and applying the identities mentioned above.