Lecture Notes: Trigonometric Identities and Equations

Overview

• Focus on exact trig values, the unit circle, and the CAST diagram.
• Key identities: ( \sin/\cos = \tan ), ( \sin^2 + \cos^2 = 1 ).
• Solving trig equations and using quadratic forms.

Unit Circle

• Theta Range (0 to 90):
  • Sine ( \theta ): Length of the opposite side of a right triangle.
  • Cosine ( \theta ): Length of the adjacent side of a right triangle.
  • Hypotenuse: 1 in the unit circle.
  • Gradient of bold line: ( \frac{\sin x}{\cos x} = \tan x ).

CAST Diagram

• Measuring angles anti-clockwise.
• Quadrants:
  • 0 to 90: All positive.
  • 90 to 180: Only sine is positive.
  • 180 to 270: Only tangent is positive.
  • 270 to 360: Only cosine is positive.

Graphs of Trigonometric Functions

• Sine and Cosine Graphs:
  • Periodic, with max of 1 and min of -1.
  • Translation relationship: Cosine is a shifted sine graph.
• Tangent Graph:
  • Periodic with no max/min.
  • Asymptotes present.

Using CAST Diagram

• Helps identify angles with positive values in specific quadrants.
• Solving trigonometric equations using known values and transformations.

Trigonometric Identities

• Pythagorean Identity:
  • ( \sin^2 \theta + \cos^2 \theta = 1 ).
• Tangent Identity:
  • ( \tan \theta = \frac{\sin \theta}{\cos \theta} ).
• Proofs:
  • Based on manipulating known identities to arrive at new equations.

Solving Trigonometric Equations with CAST

• Identify angle within given interval using CAST.
• Reflect angles appropriately depending on sine, cosine, or tangent.

Quadratic Trigonometric Equations

• Use substitution methods.
• Factorize or solve using identities.
• Example: ( 5 \sin^2 x + 3 \sin x - 2 = 0 ).

Graph Transformations

• Understanding how transformations affect solutions:
  • ( \cos(3x) ) alters the period of the cosine graph.
  • Adjust intervals and solve accordingly.

Example Problems

• Solve ( \tan^2 \theta = 4 ): Derive ( \tan \theta = \pm 2 ), find solutions.
• Address transformations in complex problems using CAST.

Practice Problems

• Various exercises to practice solving equations using the discussed methods.

Tips

• Be thorough with CAST setup and interval adjustments.
• Always double-check against initial conditions and intervals.
• Use symmetry and known values effectively.