Trigonometry

  1. Solving Trigonometric Equations
    1. Basic Trigonometric Equations
      1. Linear Equations in Trigonometric Functions
        1. sin x = a
          1. cos x = a
            1. tan x = a
            2. Using Inverse Functions
              1. Finding Principal Solutions
                1. Calculator Techniques
                2. Finding All Solutions in Given Interval
                  1. Using Unit Circle
                    1. Reference Angle Method
                      1. Periodic Nature Consideration
                    2. Algebraic Techniques
                      1. Factoring Trigonometric Expressions
                        1. Common Factor Extraction
                          1. Grouping Methods
                            1. Special Factoring Patterns
                            2. Quadratic Equations in Trigonometric Functions
                              1. Standard Quadratic Form
                                1. Using Quadratic Formula
                                  1. Factoring Quadratic Expressions
                                  2. Substitution Methods
                                    1. Single Variable Substitution
                                      1. Using Trigonometric Identities
                                        1. Simplifying Complex Equations
                                      2. Equations with Multiple Angles
                                        1. Double-Angle Equations
                                          1. sin(2x) = a
                                            1. cos(2x) = a
                                              1. tan(2x) = a
                                              2. Half-Angle Equations
                                                1. Solving for x/2 First
                                                  1. Adjusting Final Solutions
                                                  2. General Multiple-Angle Cases
                                                    1. sin(nx) = a
                                                      1. Period Considerations
                                                    2. Using Trigonometric Identities
                                                      1. Pythagorean Identity Applications
                                                        1. Converting Between sin and cos
                                                          1. Eliminating Variables
                                                          2. Sum and Difference Formula Applications
                                                            1. Expanding Compound Angles
                                                              1. Simplifying Expressions
                                                              2. Double-Angle and Half-Angle Applications
                                                                1. Reducing Equation Complexity
                                                                  1. Standard Form Conversion
                                                                2. General Solutions
                                                                  1. Expressing Solutions with Parameters
                                                                    1. Using n as Integer Parameter
                                                                      1. Degree vs Radian Notation
                                                                      2. Solution Families
                                                                        1. sin x = a: x = arcsin a + 2πn, x = π - arcsin a + 2πn
                                                                          1. cos x = a: x = ±arccos a + 2πn
                                                                            1. tan x = a: x = arctan a + πn
                                                                            2. Restricting to Given Intervals
                                                                              1. Finding Specific Solutions
                                                                                1. Checking Boundary Conditions
                                                                                2. Verification of Solutions
                                                                                  1. Substitution Check
                                                                                    1. Graphical Verification

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                                                                                6. Inverse Trigonometric Functions

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                                                                                8. Applications of Trigonometry