Study Notes on Inverse Functions and Trigonometry

Introduction

  • Welcome to the class, continuation of previous topics.

  • Focus on inverse functions, trigonometry, and trigonometric inverse functions.

  • Importance of preparing for the exam next week, which includes inverse functions.

Exam Preparation

  • Exam format includes questions on:

    • Graphs

    • Asymptotes

    • Domains

    • Inverse functions

  • Revision of the review material is crucial.

Inverse Functions Review

Definition of Inverse Functions

  • Inverse functions undo the operations of the original function.

  • Examples: sine (sin) and sine inverse (sin-1).

Evaluating Compositions of Functions

  • The process starts from the inside of the composition.

  • Example: Cosine inverse of sine(7π/3).

Analyzing Sine(7π/3)

  • Understanding angles through the unit circle:

    • Complete a circle in thirds:

      • 2π = 6π/3 (full circle).

      • 7π/3 is one third beyond a full circle.

  • Conversion: sine(7π/3) = sine(π/3).

  • Value: sine(π/3) = √3/2 (from the unit circle).

Final Steps for Cosine Inverse

  • Now evaluate: cosine inverse(√3/2).

  • Identify angles where cosine equals √3/2:

    • Possible angles: π/6, 11π/6.

  • Range for cosine inverse is [0, π], hence:

    • Only π/6 is valid within the range.

Summary of Trigonometric Values

  • Cosine inverse(√3/2) = π/6.

  • Evaluation of negative one in cosine:

    • Examples of sine and its range.

Common Error Discussion

  • Examined the value of sine inverse(-1) and found:

    • Sine inverse of -1 leads to 3π/2 (incorrect as it is out of range).

    • Corrected to find an equivalent angle within its valid range, which must fall between -π/2 to π/2.

Inverse Function Analysis

Example: Tangent Inverse of Cosine(2π/5)

  • Establish if value is from the unit circle:

    • Tangent range [-π/2, π/2], therefore possible.

  • Use calculator to evaluate cosine of 2π/5:

    • Approximate value: 0.2997.

  • Outline of calculating values leveraging calculators.

Applying the Concepts

  • Problem-solving and evaluating ranges of angles is essential.

  • Review previous examinations of inverse function components.

Example Problem Solutions

  1. Determining sin(2π/3):

    • Obtained value of √3/2.

    • Relevant for examination topics.

  2. Tangent inverse calculations:

    • Need for calculator for angles not in the unit circle.

Supplementary Exercises

  • Students should practice problems and prepare for subsequent classes.

  • Review inverse functions and associated graphs.

  • Emphasis on identifying ranges and quadrant considerations.

Final Reminders

  • Utilize inverse functions for associated sine, cosine, and tangent relationships effectively.

  • Engage with class materials and reach out for clarifications if needed.

  • Homework assignments should reinforce these concepts to solidify understanding for the exam preparation.