Trigonometry

  1. Trigonometric Identities
    1. Fundamental Identities
      1. Reciprocal Identities
        1. sin θ · csc θ = 1
          1. cos θ · sec θ = 1
            1. tan θ · cot θ = 1
              1. Alternative Forms
              2. Quotient Identities
                1. tan θ = sin θ / cos θ
                  1. cot θ = cos θ / sin θ
                    1. Derivation from Definitions
                    2. Pythagorean Identities
                      1. sin²θ + cos²θ = 1
                        1. Derivation from Unit Circle
                          1. Geometric Interpretation
                          2. 1 + tan²θ = sec²θ
                            1. Derivation from Basic Identity
                              1. Alternative Forms
                              2. 1 + cot²θ = csc²θ
                                1. Derivation Process
                            2. Verifying Trigonometric Identities
                              1. Strategies for Verification
                                1. Work with More Complex Side
                                  1. Use Fundamental Identities
                                    1. Factor When Possible
                                      1. Find Common Denominators
                                      2. Common Techniques
                                        1. Substitution Using Basic Identities
                                          1. Multiplying by Conjugates
                                            1. Converting to Sine and Cosine
                                            2. Working from One Side
                                              1. Left-to-Right Approach
                                                1. Right-to-Left Approach
                                                  1. Meeting in the Middle
                                                  2. Common Pitfalls
                                                    1. Treating Identity as Equation
                                                      1. Working Both Sides Simultaneously
                                                        1. Circular Reasoning
                                                      2. Sum and Difference Formulas
                                                        1. Sine Sum and Difference
                                                          1. sin(A + B) = sin A cos B + cos A sin B
                                                            1. sin(A - B) = sin A cos B - cos A sin B
                                                              1. Derivation Methods
                                                                1. Memory Techniques
                                                                2. Cosine Sum and Difference
                                                                  1. cos(A + B) = cos A cos B - sin A sin B
                                                                    1. cos(A - B) = cos A cos B + sin A sin B
                                                                      1. Geometric Derivation
                                                                      2. Tangent Sum and Difference
                                                                        1. tan(A + B) = (tan A + tan B)/(1 - tan A tan B)
                                                                          1. tan(A - B) = (tan A - tan B)/(1 + tan A tan B)
                                                                            1. Derivation from Sine and Cosine Formulas
                                                                            2. Applications of Sum and Difference Formulas
                                                                              1. Exact Values for Non-Standard Angles
                                                                                1. Simplifying Complex Expressions
                                                                                  1. Solving Equations
                                                                                2. Double-Angle Formulas
                                                                                  1. Sine Double-Angle
                                                                                    1. sin(2A) = 2 sin A cos A
                                                                                      1. Derivation from Sum Formula
                                                                                      2. Cosine Double-Angle
                                                                                        1. cos(2A) = cos²A - sin²A
                                                                                          1. cos(2A) = 2cos²A - 1
                                                                                            1. cos(2A) = 1 - 2sin²A
                                                                                              1. Multiple Forms and Uses
                                                                                              2. Tangent Double-Angle
                                                                                                1. tan(2A) = 2tan A/(1 - tan²A)
                                                                                                  1. Domain Restrictions
                                                                                                  2. Applications of Double-Angle Formulas
                                                                                                    1. Simplifying Expressions
                                                                                                      1. Integration Techniques
                                                                                                        1. Solving Equations
                                                                                                      2. Power-Reducing Formulas
                                                                                                        1. Derivation from Double-Angle Formulas
                                                                                                          1. Solving for sin²A and cos²A
                                                                                                          2. Power-Reducing Formulas
                                                                                                            1. sin²A = (1 - cos(2A))/2
                                                                                                              1. cos²A = (1 + cos(2A))/2
                                                                                                                1. tan²A = (1 - cos(2A))/(1 + cos(2A))
                                                                                                                2. Applications
                                                                                                                  1. Integration of Powers
                                                                                                                    1. Simplifying Complex Expressions
                                                                                                                  2. Half-Angle Formulas
                                                                                                                    1. Sine Half-Angle
                                                                                                                      1. sin(A/2) = ±√[(1 - cos A)/2]
                                                                                                                        1. Sign Determination
                                                                                                                        2. Cosine Half-Angle
                                                                                                                          1. cos(A/2) = ±√[(1 + cos A)/2]
                                                                                                                            1. Sign Determination
                                                                                                                            2. Tangent Half-Angle
                                                                                                                              1. tan(A/2) = ±√[(1 - cos A)/(1 + cos A)]
                                                                                                                                1. Alternative Forms
                                                                                                                                  1. tan(A/2) = sin A/(1 + cos A)
                                                                                                                                    1. tan(A/2) = (1 - cos A)/sin A
                                                                                                                                    2. Applications of Half-Angle Formulas
                                                                                                                                      1. Exact Values
                                                                                                                                        1. Integration Techniques
                                                                                                                                      2. Product-to-Sum and Sum-to-Product Formulas
                                                                                                                                        1. Product-to-Sum Formulas
                                                                                                                                          1. sin A cos B = ½[sin(A + B) + sin(A - B)]
                                                                                                                                            1. cos A sin B = ½[sin(A + B) - sin(A - B)]
                                                                                                                                              1. cos A cos B = ½[cos(A + B) + cos(A - B)]
                                                                                                                                                1. sin A sin B = ½[cos(A - B) - cos(A + B)]
                                                                                                                                                2. Sum-to-Product Formulas
                                                                                                                                                  1. sin A + sin B = 2sin[(A + B)/2]cos[(A - B)/2]
                                                                                                                                                    1. sin A - sin B = 2cos[(A + B)/2]sin[(A - B)/2]
                                                                                                                                                      1. cos A + cos B = 2cos[(A + B)/2]cos[(A - B)/2]
                                                                                                                                                        1. cos A - cos B = -2sin[(A + B)/2]sin[(A - B)/2]
                                                                                                                                                        2. Applications
                                                                                                                                                          1. Simplifying Products and Sums
                                                                                                                                                            1. Solving Equations
                                                                                                                                                              1. Harmonic Analysis

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                                                                                                                                                          4. Graphs of Trigonometric Functions

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                                                                                                                                                          6. Inverse Trigonometric Functions