- Ch. 1The Unit Circle and the Trigonometric FunctionsTrig functions are the coordinates of a point on the unit circle.
- Ch. 1Graphs and Their TransformationsA sin(ωx+φ)+k is a shifted, stretched sine; the shift depends on the order.
- Ch. 1Period, Monotonicity and SymmetryShift or mirror a copy of the graph; if it fits, you found a period or symmetry.
- Ch. 2Sum, Double-Angle and Half-Angle FormulasStack two right triangles to get sin(α+β); the other formulas are special cases.
- Ch. 2Sum-to-Product and Product-to-SumAdd two unit vectors and read the diagonal twice: a sum becomes a product.
- Ch. 2The Auxiliary Angle and Evaluating Expressionsa sin x + b cos x is the height of a turning L with reach √(a²+b²).
- Ch. 3The Laws of Sines and CosinesSide = diameter × sine of the opposite angle; Pythagoras plus −2ab cos C.
- Ch. 3Identities and Area in a TriangleOnly two angles are free: drop one with C = π − (A+B) and identities follow.
- Ch. 4Inverse Trigonometric FunctionsSine is not one-to-one. Cut out a monotone piece and reflect it in y = x.
- Ch. 4Simple Trigonometric EquationsReduce to sin x = a, solve on one turn of the circle, then add the period.
- Ch. 5Trigonometric InequalitiesSolve with graphs and the circle; prove with area, monotonicity and concavity.
- Ch. 6Trigonometric Substitution and ApplicationsSee x²+y²=1, √(1−x²) or 2x/(1+x²)? Move it onto the unit circle.