Lecture Notes: Inverse Trig Functions

Introduction

• Inverse trig functions are used to find angles when given a ratio of sides in a right triangle.
• This lecture aims to deepen the understanding of inverse trig functions, moving beyond basic geometry.

Quick Review of Trig Functions

• Trig functions relate angles to side ratios:
  • Sine (sin): Opposite over hypotenuse.
  • Example: sin(42°) gives a ratio (e.g., 0.6691), which is the opposite side over hypotenuse for any right triangle with a 42° angle.

Inverse Trig Functions

• Inverse trig functions are used to find the angle when given the ratio.
• Notation: sin⁻¹(x) or arcsin(x).
• Example: To find angle [3/8], use inverse sine (sin⁻¹).

Graphing Inverse Trig Functions

• Regular Sine Function (s of x):
  • Domain: All real numbers (ℝ).
  • Range: [-1, 1].
• Inverses:
  • Switch x and y: Domain becomes range and vice versa.
  • Inverse sine (arcsin) is graphed by limiting domain from -π/2 to π/2.

Principal Values

• Inverse trig functions are restricted to principal values to make them functions:
  • Sine & Tangent: Quadrants 1 and 4.
  • Cosine: Quadrants 1 and 2.

Graphical Representation

• Inverse sine (arcsin) is mapped from -π/2 to π/2.
• Inverse cosine (arccos) is mapped from 0 to π.
• Inverse tangent (arctan) maps from -π/2 to π/2.

Solving Equations with Inverse Trig Functions

• Use inverse trig to solve for angles in equations. Example process:
  1. Switch x and y.
  2. Solve for y (apply inverse trig).

Applications with Unit Circle

• Unit Circle: Helps in finding principal values.
• Example: Inverse sin(1/2) corresponds to angle π/6 in quadrant 1.
• Cosine and Tangent: Similar process, observing quadrant restrictions.

Problem-Solving Tips

• Always ensure calculator is in correct mode (radians vs. degrees).
• Pay attention to domain and range restrictions for each inverse trig function.

Conclusion

• Mastery of inverse trig functions is essential for solving more complex trig equations and is integral to further studies in mathematics.
• Practice is key to familiarity and fluency in using these functions.