5.4 Inverse Trigonometric Functions-20260306_105706-Meeting Recording

Introduction

  • Objective: Learn about inverse trigonometric functions.
  • Context: Previously used inverse trig functions without in-depth understanding. Today’s lesson focuses on their specific properties and calculations.

Review of Cosine Function

  • Previous Work: Solving problems like ( \cos(60^{\circ}) = ? )
  • Understanding: Used the unit circle to find values.
    • For ( 60^{\circ} ): X-coordinate is ( \frac{1}{2} ).

Introduction to Inverse Trig Functions

  • Problem Presented: Finding angle ( \theta ) given a Y-coordinate.
  • Example Given: Determine where Y-coordinate equals ( -\frac{\sqrt{3}}{2} ).
  • Solutions Found: ( 240^{\circ} ) and ( 300^{\circ} ).

Understanding the Inverse Operation

  • Previous Lesson: Learned about solving equations with inverse operations.
  • Inverse Operation: ( \sin^{-1}(x) ) is used to "undo" the sine function.
    • Example: To find ( \theta ), apply ( \sin^{-1} ) to both sides.

Calculating Inverse Sine

  • Calculator Practice: Compute ( \sin^{-1}(-\frac{\sqrt{3}}{2}) ).
  • Calculator Output: ( -60^{\circ} )
    • Explanation of Surprise: Negative angle reflects a coterminal angle with ( 300^{\circ} ).

Properties of Inverse Functions

  • Understanding the Inverse Function Graph:
    • Example Function: ( y = x^{2} ) is not a one-to-one function.
    • Inverse: Requires modifying the function to pass the horizontal line test.
    • Procedure: Restrict the domain to keep it one-to-one.

Inverse of Sine Function

  • Graph Characteristics:
    • The sine function does not pass the horizontal line test.
    • To find the inverse, restrict the graph from ( -90^{\circ} ) to ( 90^{\circ} ) (or ( -\frac{\pi}{2} ) to ( \frac{\pi}{2} )).
  • Confusion Cleared: Why does the calculator give negative output instead of larger angles? Only outputs angles within the specified range.

Output Requirements for Inverse Trig Functions

  • Important Results:
    • Inverse Sine outputs values only between ( -90^{\circ} ) and ( 90^{\circ} ).
    • Use quadrants 1 (positive values) and negative quadrant 1 (negative values).

Inverse Cosine Function

  • Restricted Output: Between ( 0^{\circ} ) and ( 180^{\circ} ) (or ( 0 ) to ( \pi )).
    • Quadrants: 1 (positive values), 2 (negative values).
  • Example Calculation: ( \cos^{-1}(-\frac{1}{2}) ) leads to solutions in quadrant 2.
  • Problem Calculated: Find ( \cos^{-1}(\frac{\sqrt{2}}{2}) ); output is ( 45^{\circ} ).

Inverse Tangent Function

  • Characteristics:
    • Restricted Output: Between ( -90^{\circ} ) and ( 90^{\circ} ).
    • Cannot reach ( 90^{\circ} ) or ( -90^{\circ} ); output remains within the defined constraints.
  • Understanding Tangent: Defined as ( \tan(\theta) = \frac{y}{x} ).
    • Highlights that tangent at ( 90^{\circ} ) is undefined due to division by zero.
  • Example Discussed:
    • Condition required to be the same: both X and Y coordinates must match for tangents to equal 1.

Problem-Solving Using Inverse Functions

  • Ladder Problem: Involves inverse sine for triangle formed by ladder against a wall.
    • Given: Ladder (hypotenuse) = 40ft, distance from wall = 6ft.
    • Equation Formed: ( \sin(\theta) = \frac{6}{40} ) leads to ( \theta = \sin^{-1}(\frac{6}{40}) ). Output approximately ( 8.6^{\circ} ).

Practice Problems

  1. Find all angles ( \theta ) for which ( \sin(\theta) = -\frac{24}{7} ).
  2. Understand output must conform to calculator's