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Unit 3 · Trigonometric & Polar Functions Flashcards Cheat Sheet Essentials Visual Review MC Practice FRQ Practice

AP Precalculus Unit 3 Essentials

The must-know terms and core concepts for Unit 3: Trigonometric & Polar Functions. Every vocabulary word and idea you need to master.

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Key Concept 1
Trigonometric functions come from the unit circle and model periodic phenomena
For an angle θ in standard position, the point where its terminal ray meets the unit circle is (cos θ, sin θ), and tangent is their ratio. Because those coordinates repeat with every full revolution, the trig functions are periodic — ideal for modeling anything that repeats. Reading a periodic graph means identifying its period (length of one cycle), amplitude (half the max−min distance), and midline.
Unit Circle Periodic Phenomena Sine & Cosine
Key Concept 2
Sinusoidal functions are transformations, and periodicity shapes their equations
A sinusoidal function has the form f(x) = a·sin(b(x − c)) + d, where a controls amplitude, b sets the period (2π/|b|), c is the phase shift, and d is the midline — the same transformation logic from Unit 1 applied to a periodic parent. Because trig functions repeat, inverse trig functions require restricted domains to be one-to-one, and trigonometric equations have infinitely many solutions, found by adding integer multiples of the period. Identities such as sin²θ + cos²θ = 1 let you rewrite expressions in equivalent forms.
Sinusoidal Functions Inverse Trig Trig Equations
Key Concept 3
Polar coordinates give a second way to locate points and graph functions
The polar system describes a point as (r, θ) — a directed distance and an angle — and connects to rectangular coordinates through x = r·cos θ and y = r·sin θ. Graphing r = f(θ) produces distinctive curves (circles, roses, limaçons, cardioids, spirals), and the rate of change of r with respect to θ tells you where the curve moves toward or away from the pole.
Polar Coordinates Polar Graphs Rates of Change
Periodic function
A function whose values repeat at regular intervals; the period is the smallest such interval.
Periodic
Period
The length of one complete cycle of a periodic function; for a sinusoid it equals 2π/|b|.
Periodic
Amplitude
Half the distance between the maximum and minimum values of a sinusoidal function; equal to |a|.
Periodic
Midline
The horizontal line halfway between the maximum and minimum, y = d; the value a sinusoid oscillates around.
Periodic
Radian
An angle measure based on arc length: an arc equal to the radius subtends 1 radian, and a full circle is 2π radians.
Unit Circle
Unit circle
The circle of radius 1 centered at the origin; for angle θ the point on it is (cos θ, sin θ).
Unit Circle
Sine and cosine
The y- and x-coordinates, respectively, of the unit-circle point for an angle θ. Domain all reals, range from −1 to 1.
Unit Circle
Tangent
tan θ = sin θ / cos θ; period π, with vertical asymptotes wherever cos θ = 0.
Unit Circle
Sinusoidal function
A function of the form f(x) = a·sin(b(x − c)) + d (or with cosine), modeling smooth periodic behavior.
Sinusoidal
Phase shift
The horizontal shift c of a sinusoidal graph.
Sinusoidal
Frequency
The number of cycles per unit of input; it is the reciprocal of the period.
Sinusoidal
Inverse trigonometric function
arcsin, arccos, or arctan — the inverse of a trig function on a restricted, one-to-one domain.
Inverse & Equations
Restricted domain
A limited set of inputs on which a trig function is one-to-one, so that an inverse can be defined.
Inverse & Equations
Trigonometric equation
An equation involving a trig function; because of periodicity it usually has infinitely many solutions.
Inverse & Equations
Secant, cosecant, cotangent
The reciprocal functions: sec = 1/cos, csc = 1/sin, cot = cos/sin.
Reciprocals & Identities
Pythagorean identity
sin²θ + cos²θ = 1, from which 1 + tan²θ = sec²θ follows.
Reciprocals & Identities
Even / odd functions
Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
Reciprocals & Identities
Polar coordinates
A point given as (r, θ): a directed distance r from the pole and an angle θ from the polar axis.
Polar
Pole and polar axis
The origin of the polar system (the pole) and the reference ray from which θ is measured (the polar axis).
Polar
Polar–rectangular conversion
x = r cos θ and y = r sin θ; r = √(x² + y²) and tan θ = y/x.
Polar
Polar graph
The graph of r = f(θ), producing curves such as circles, roses, limaçons, cardioids, and spirals.
Polar
Rate of change in polar
The change in r with respect to θ, describing where a polar curve nears or leaves the pole.
Polar