Trigonometric and Polar Functions
Fifteen topics on periodic change. The unit circle as the engine behind sine, cosine, and tangent, sinusoidal functions with amplitude, midline, period, and phase, transformations and context modeling, the tangent and reciprocal functions, inverse trig with its restricted ranges, equation solving that keeps every solution, and polar coordinates with the curves they draw.
Topics
Key forms For every problem in this unit
General sinusoid
f(x) = a·sin(b(x − h)) + k
Amplitude
|a| (distance midline to peak, never negative)
Period
2π/|b| for sin and cos; π/|b| for tan
Midline
y = k
Phase shift
h, read only AFTER factoring b out of the argument
Pythagorean identity
sin²θ + cos²θ = 1
Tangent
tan θ = sin θ / cos θ (undefined where cos θ = 0)
Polar to rectangular
x = r·cos θ, y = r·sin θ
Rectangular to polar
r² = x² + y², tan θ = y/x (quadrant-checked)
Inverse trig ranges
arcsin: [−π/2, π/2] · arccos: [0, π] · arctan: (−π/2, π/2)
Unit 3 tools