Lecture on Trigonometric Functions and Their Inverses

Key Concepts

Trigonometric Functions

• Right Triangle Trigonometry:
  • Ratios depend on the central angle θ, not on the triangle size.
  • Common Ratios:
    • Sine (sin θ): Opposite / Hypotenuse
    • Cosine (cos θ): Adjacent / Hypotenuse
    • Tangent (tan θ): Opposite / Adjacent
    • Reciprocal Ratios:
      • Cosecant (csc θ): 1/sin θ
      • Secant (sec θ): 1/cos θ
      • Cotangent (cot θ): 1/tan θ

Unit Circle

• Gives values of sin, cos, and consequently tan for common angles.
• Important to memorize one quadrant for sine and cosine values.
• Quadrants:
  • Quadrant 1: All positive
  • Quadrant 2: Sin positive, cos negative
  • Quadrant 3: All negative
  • Quadrant 4: Cos positive, sin negative

Calculating Trigonometric Values

• Example: Given sin θ = 3/5, find cos θ and tan θ.
  • Use Pythagorean theorem to find missing side.
  • Adjust signs based on the quadrant.

Trigonometric Identities

• Pythagorean Identity: sin²θ + cos²θ = 1
• Other identities derived by manipulating Pythagorean identity.
• Functions are periodic and not one-to-one.
  • Period: Value where the function repeats its outputs.

Inverse Trigonometric Functions

Concept

• Trig functions take an angle and return a ratio.
• Inverse functions take a ratio and return an angle.
• Normally undefined for non-one-to-one functions, so restrict domain to make them one-to-one.

Domain and Range

• Inverse Sine (arcsin):
  • Domain: -1 to 1
  • Range: -π/2 to π/2
• Inverse Cosine (arccos):
  • Domain: -1 to 1
  • Range: 0 to π
• Inverse Tangent (arctan):
  • Open interval -π/2 to π/2

Graphical Representation

• Reflect the restricted domain graph over y = x to get inverse functions.

Solving Trigonometric Equations

Quadratic-like Trigonometric Equations

• Convert trigonometric expressions to solve as quadratics.
• Use factoring techniques and unit circle knowledge to find solutions.

Practical Applications

Evaluating Expressions

• Example: Evaluate arc cosine of a given ratio.
• Example: Evaluate cosine of the arc cosine of a value.
  • Note: Ensure angles lie within the restricted range.

Using Right Triangles for Simplification

• Define an angle using inverse trig to simplify trigonometric expressions.

Important Tips

• Memorize unit circle values.
• Understand domain restrictions for inverse trigonometric functions.
• Practice solving trigonometric equations with different methods to enhance proficiency.