Trigonometry

  1. Right Triangle Trigonometry
    1. The Six Trigonometric Ratios
      1. Sine (sin)
        1. Definition in Right Triangle
          1. Opposite over Hypotenuse
            1. Ratio Interpretation
            2. Properties of Sine
              1. Range from 0 to 1
                1. Increasing Function in First Quadrant
              2. Cosine (cos)
                1. Definition in Right Triangle
                  1. Adjacent over Hypotenuse
                    1. Ratio Interpretation
                    2. Properties of Cosine
                      1. Range from 0 to 1
                        1. Decreasing Function in First Quadrant
                      2. Tangent (tan)
                        1. Definition in Right Triangle
                          1. Opposite over Adjacent
                            1. Ratio Interpretation
                            2. Properties of Tangent
                              1. Can Take Any Real Value
                                1. Increasing Function in First Quadrant
                              2. Cosecant (csc)
                                1. Definition in Right Triangle
                                  1. Hypotenuse over Opposite
                                    1. Reciprocal of Sine
                                    2. Properties of Cosecant
                                      1. Always Greater than or Equal to 1
                                        1. Undefined When Sine is Zero
                                      2. Secant (sec)
                                        1. Definition in Right Triangle
                                          1. Hypotenuse over Adjacent
                                            1. Reciprocal of Cosine
                                            2. Properties of Secant
                                              1. Always Greater than or Equal to 1
                                                1. Undefined When Cosine is Zero
                                              2. Cotangent (cot)
                                                1. Definition in Right Triangle
                                                  1. Adjacent over Opposite
                                                    1. Reciprocal of Tangent
                                                    2. Properties of Cotangent
                                                      1. Can Take Any Real Value
                                                        1. Decreasing Function in First Quadrant
                                                    3. The SOH CAH TOA Mnemonic
                                                      1. SOH (Sine = Opposite/Hypotenuse)
                                                        1. CAH (Cosine = Adjacent/Hypotenuse)
                                                          1. TOA (Tangent = Opposite/Adjacent)
                                                            1. Using the Mnemonic for Problem Solving
                                                              1. Alternative Memory Devices
                                                              2. Special Right Triangles
                                                                1. 45-45-90 Triangles
                                                                  1. Angle Measures
                                                                    1. Side Ratios
                                                                      1. 1 : 1 : √2
                                                                      2. Trigonometric Values
                                                                        1. sin 45° = cos 45° = √2/2
                                                                          1. tan 45° = 1
                                                                          2. Applications of 45-45-90 Triangles
                                                                          3. 30-60-90 Triangles
                                                                            1. Angle Measures
                                                                              1. Side Ratios
                                                                                1. 1 : √3 : 2
                                                                                2. Trigonometric Values
                                                                                  1. Values for 30° and 60°
                                                                                  2. Applications of 30-60-90 Triangles
                                                                                  3. Deriving Special Triangle Values
                                                                                    1. Geometric Construction Methods
                                                                                      1. Algebraic Verification
                                                                                    2. Solving Right Triangles
                                                                                      1. Identifying Given Information
                                                                                        1. Known Sides and Angles
                                                                                          1. Unknown Quantities
                                                                                            1. Choosing Appropriate Strategy
                                                                                            2. Finding Unknown Side Lengths
                                                                                              1. Using Trigonometric Ratios
                                                                                                1. Selecting Appropriate Ratio
                                                                                                  1. Setting Up Equations
                                                                                                    1. Solving for Unknown
                                                                                                    2. Using Pythagorean Theorem
                                                                                                      1. When to Apply
                                                                                                        1. Verification Method
                                                                                                      2. Finding Unknown Angle Measures
                                                                                                        1. Using Inverse Trigonometric Functions
                                                                                                          1. Selecting Appropriate Inverse Function
                                                                                                            1. Calculator Usage
                                                                                                            2. Angle Sum Property
                                                                                                              1. Sum Equals 90° for Acute Angles
                                                                                                              2. Checking Solutions for Reasonableness
                                                                                                                1. Angle Size Verification
                                                                                                                  1. Triangle Inequality
                                                                                                              3. Introduction to Inverse Trigonometric Functions
                                                                                                                1. Need for Inverse Functions
                                                                                                                  1. Finding Angles from Ratios
                                                                                                                    1. Reversing Trigonometric Operations
                                                                                                                    2. Arcsin (sin⁻¹)
                                                                                                                      1. Definition and Notation
                                                                                                                        1. Domain and Range
                                                                                                                          1. Domain: [-1, 1]
                                                                                                                            1. Range: [-90°, 90°]
                                                                                                                            2. Evaluating Arcsin
                                                                                                                              1. Calculator Methods
                                                                                                                                1. Special Values
                                                                                                                              2. Arccos (cos⁻¹)
                                                                                                                                1. Definition and Notation
                                                                                                                                  1. Domain and Range
                                                                                                                                    1. Domain: [-1, 1]
                                                                                                                                      1. Range: [0°, 180°]
                                                                                                                                      2. Evaluating Arccos
                                                                                                                                        1. Calculator Methods
                                                                                                                                          1. Special Values
                                                                                                                                        2. Arctan (tan⁻¹)
                                                                                                                                          1. Definition and Notation
                                                                                                                                            1. Domain and Range
                                                                                                                                              1. Domain: All Real Numbers
                                                                                                                                                1. Range: (-90°, 90°)
                                                                                                                                                2. Evaluating Arctan
                                                                                                                                                  1. Calculator Methods
                                                                                                                                                    1. Special Values
                                                                                                                                                3. Applications of Right Triangle Trigonometry
                                                                                                                                                  1. Angle of Elevation
                                                                                                                                                    1. Definition of Angle of Elevation
                                                                                                                                                      1. Setting Up Problems
                                                                                                                                                        1. Identifying Observer and Object
                                                                                                                                                          1. Drawing Diagrams
                                                                                                                                                          2. Solving Elevation Problems
                                                                                                                                                            1. Height Calculations
                                                                                                                                                              1. Distance Calculations
                                                                                                                                                            2. Angle of Depression
                                                                                                                                                              1. Definition of Angle of Depression
                                                                                                                                                                1. Relationship to Angle of Elevation
                                                                                                                                                                  1. Setting Up Problems
                                                                                                                                                                    1. Solving Depression Problems

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                                                                                                                                                                1. Foundations of Trigonometry

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                                                                                                                                                                3. Trigonometric Functions of Any Angle