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.