Trigonometry

  1. Inverse Trigonometric Functions
    1. Review of Inverse Functions
      1. Definition of Inverse Function
        1. Reversing Input and Output
          1. Notation Conventions
          2. One-to-One Function Requirement
            1. Horizontal Line Test
              1. Need for Domain Restrictions
              2. Relationship Between Function and Inverse
                1. Composition Properties
                  1. Graphical Relationship
                2. Domain and Range Restrictions
                  1. Need for Principal Values
                    1. Multiple Angle Solutions
                      1. Choosing Standard Intervals
                      2. Inverse Sine Function
                        1. Domain: [-1, 1]
                          1. Range: [-π/2, π/2]
                            1. Principal Value Branch
                              1. Notation: arcsin x or sin⁻¹ x
                              2. Inverse Cosine Function
                                1. Domain: [-1, 1]
                                  1. Range: [0, π]
                                    1. Principal Value Branch
                                      1. Notation: arccos x or cos⁻¹ x
                                      2. Inverse Tangent Function
                                        1. Domain: All Real Numbers
                                          1. Range: (-π/2, π/2)
                                            1. Principal Value Branch
                                              1. Notation: arctan x or tan⁻¹ x
                                              2. Other Inverse Functions
                                                1. Inverse Cotangent
                                                  1. Domain and Range
                                                  2. Inverse Secant
                                                    1. Domain and Range
                                                      1. Convention Variations
                                                      2. Inverse Cosecant
                                                        1. Domain and Range
                                                    2. Graphs of Inverse Trigonometric Functions
                                                      1. Graph of y = arcsin x
                                                        1. Domain and Range Visualization
                                                          1. Key Points
                                                            1. Increasing Function
                                                              1. Horizontal Asymptotes
                                                              2. Graph of y = arccos x
                                                                1. Domain and Range Visualization
                                                                  1. Key Points
                                                                    1. Decreasing Function
                                                                    2. Graph of y = arctan x
                                                                      1. Domain and Range Visualization
                                                                        1. Horizontal Asymptotes at ±π/2
                                                                          1. Increasing Function
                                                                            1. Point of Inflection
                                                                            2. Relationship to Original Functions
                                                                              1. Reflection Across y = x
                                                                                1. Domain-Range Interchange
                                                                              2. Evaluating Inverse Trigonometric Functions
                                                                                1. Exact Values for Special Angles
                                                                                  1. Common Values to Memorize
                                                                                    1. Using Reference Triangles
                                                                                    2. Calculator Usage
                                                                                      1. Mode Settings
                                                                                        1. Interpreting Results
                                                                                        2. Compositions with Trigonometric Functions
                                                                                          1. sin(arcsin x) = x
                                                                                            1. arcsin(sin x) considerations
                                                                                              1. Domain Restrictions Impact
                                                                                            2. Compositions and Complex Expressions
                                                                                              1. Evaluating sin(arccos x)
                                                                                                1. Right Triangle Method
                                                                                                  1. Algebraic Method
                                                                                                  2. Evaluating tan(arcsin x)
                                                                                                    1. Triangle Construction
                                                                                                      1. Pythagorean Theorem Application
                                                                                                      2. General Strategy for Compositions
                                                                                                        1. Drawing Reference Triangles
                                                                                                          1. Using Fundamental Identities

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                                                                                                      5. Trigonometric Identities

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                                                                                                      7. Solving Trigonometric Equations