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

AP Precalculus Unit 3 Visual Review

A topic-by-topic visual walkthrough of Unit 3: Trigonometric and Polar Functions — the unit circle, sinusoidal functions, inverse trig, trig equations, and polar graphs.

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TOPIC 3.1 Periodic Phenomena one period max Repeating patterns A PERIODIC function repeats its values at regular intervals: f(x + P) = f(x). PERIOD P = length of one full cycle. Amplitude & midline amplitude = (max − min) / 2 midline y = (max + min) / 2 Examples: tides, sound, seasons, rotation. Periodic: f(x+P)=f(x); amplitude=(max−min)/2, midline=(max+min)/2. The Review Hub · AP Precalculus Unit 3 TOPIC 3.2 Sine, Cosine, and Tangent (cos θ, sin θ) sincos θ cos θ = x, sin θ = y tan θ = sin θ / cos θ = y / x On the unit circle (radius 1) A point at angle θ has coordinates (cos θ, sin θ). Angles measured in RADIANS: 180° = π radians. sin²θ + cos²θ = 1 (Pythagorean identity) On the unit circle: cos θ = x, sin θ = y, tan θ = y/x. The Review Hub · AP Precalculus Unit 3 TOPIC 3.3 Sine and Cosine Function Values θ0π/6π/4π/3π/2 sin01/2√2/2√3/21 cos1√3/2√2/21/20 tan0√3/31√3undef Use symmetry & reference angles for other quadrants Signs by quadrant (ASTC): all +, then sin +, then tan +, then cos +. e.g. cos(2π/3) = −1/2, sin(2π/3)=√3/2. Memorize the special-angle values; use reference angles & ASTC signs for other quadrants. The Review Hub · AP Precalculus Unit 3 TOPIC 3.4 Sine and Cosine Function Graphs sin x cos x both have period 2π, amplitude 1; cos = sin shifted left π/2 sin x Starts at 0, odd function: sin(−x) = −sin x. cos x Starts at 1, even function: cos(−x) = cos x. 〰sin & cos have period 2π, amplitude 1; cos is sin shifted left π/2. The Review Hub · AP Precalculus Unit 3 TOPIC 3.5 Sinusoidal Functions f(x) = a · sin( b(x − h) ) + k What each parameter does |a| = AMPLITUDE (height from midline). k = MIDLINE (vertical shift), y = k. h = PHASE SHIFT (horizontal). PERIOD = 2π / |b| b compresses/stretches horizontally. Worked example f(x) = 3 sin(2x) + 5 amplitude = 3 period = 2π/2 = π midline y = 5 max = 8, min = 2 〰a=amplitude, k=midline, period=2π/|b|, h=phase shift. The Review Hub · AP Precalculus Unit 3 TOPIC 3.6 Sinusoidal Transformations Build a model from features From a graph or context, read off: a = (max − min)/2 k = (max + min)/2 b = 2π / period h from where a cycle starts. Sine ↔ cosine The same graph can be written with either sine or cosine — just a phase shift apart. sin(x) = cos(x − π/2) Reflect if a < 0 (over the midline). Order of transformations matters Apply the horizontal stretch (b) and phase shift (h) to x inside; amplitude (a) and midline (k) act on the output. Get a, k from max/min and b from the period; sine and cosine differ by a phase shift. The Review Hub · AP Precalculus Unit 3 TOPIC 3.7 Sinusoidal Context & Data Modeling Example: daylight hours Max 14h (summer), min 10h (winter): a = (14−10)/2 = 2 k = (14+10)/2 = 12 period = 365 → b = 2π/365 h(t)=2 sin(2π(t−80)/365)+12 Model periodic contexts Tides, temperatures, hours of daylight, a Ferris wheel's height, sound waves. Use sinusoidal regression to fit data, then predict future values. State the domain over which the model is valid. Interpret parameters in context Amplitude = half the swing, midline = the average level, period = time for one full cycle of the phenomenon. Model periodic data with a sinusoid — get a, k, b, h from the context's max/min/period. The Review Hub · AP Precalculus Unit 3 TOPIC 3.8 The Tangent Function x=−π/2x=π/2 tan x = sin x / cos x PERIOD = π (not 2π). Vertical asymptotes where cos x = 0, i.e. x = π/2 + nπ Behavior Increasing on each interval between asymptotes; range is all real numbers. Odd function: tan(−x) = −tan x; zero at x=0. tan x = sin x/cos x, period π, with vertical asymptotes at x = π/2 + nπ. The Review Hub · AP Precalculus Unit 3 TOPIC 3.9 Inverse Trigonometric Functions Inverse trig gives an ANGLE from a ratio arcsin, arccos, arctan undo sine, cosine, tangent — they answer "what angle has this value?" arcsin(1/2) = π/6 arctan(1) = π/4 Restricted ranges (so each is a function) arcsin x : range [ −π/2 , π/2 ] domain [−1, 1] arccos x : range [ 0 , π ] domain [−1, 1] arctan x : range ( −π/2 , π/2 ) domain all reals The calculator returns only the value in the restricted range. Inverse trig returns an angle; each has a restricted range so it's a function. The Review Hub · AP Precalculus Unit 3 TOPIC 3.10 Trigonometric Equations & Inequalities Worked example sin x = 1/2, 0 ≤ x < 2π Reference angle: arcsin(1/2) = π/6. sin > 0 in quadrants I and II: x = π/6 and x = 5π/6 Add 2πn for ALL solutions. Infinitely many solutions Because trig functions are PERIODIC, each equation has infinitely many solutions. Find them in one period, then add the period × n (2πn for sin/cos, πn for tan). Use the unit circle for the reference angle. Inequalities: use the graph Solve the equality first, then read intervals where the curve is above/below the target value. Find solutions in one period, then add the period × n for all solutions. The Review Hub · AP Precalculus Unit 3 TOPIC 3.11 Secant, Cosecant & Cotangent // reciprocal identities sec x = 1/cos x csc x = 1/sin x cot x = 1/tan x = cos x/sin x "co-" pairs: sec↔cos, csc↔sin, cot↔tan Asymptotes where the base = 0 sec/csc have vertical asymptotes where cos/sin = 0, and |value| ≥ 1 elsewhere. cot has period π like tan, but decreasing. sec/csc period 2π; ranges (−∞,−1]∪[1,∞). Example cos(π/3) = 1/2 → sec(π/3) = 1/(1/2) = 2 Take the reciprocal of the base value. sec=1/cos, csc=1/sin, cot=1/tan — asymptotes where the base equals 0. The Review Hub · AP Precalculus Unit 3 TOPIC 3.12 Equivalent Trigonometric Representations // key identities Pythagorean: sin²x + cos²x = 1 Double angle: sin 2x = 2 sin x cos x Even/odd: cos(−x)=cos x, sin(−x)=−sin x Rewrite to simplify or solve Identities let you rewrite one expression as an equivalent one that's easier to use. 1 − cos²x = sin²x Shifts as identities sin(x + π/2) = cos x cos(x − π/2) = sin x Many expressions have several equal forms. Use identities (Pythagorean, double-angle, even/odd) to rewrite trig expressions. The Review Hub · AP Precalculus Unit 3 TOPIC 3.13 Trigonometry & Polar Coordinates (r, θ) θ r x = r cos θ, y = r sin θ r = √(x²+y²), θ = arctan(y/x) convert between polar (r, θ) and rectangular (x, y) Polar coordinates Locate a point by DISTANCE r from the origin and ANGLE θ from the positive x-axis. A point has many names: (r, θ) = (r, θ + 2π). Convert with x=r cos θ, y=r sin θ and r=√(x²+y²). The Review Hub · AP Precalculus Unit 3 TOPIC 3.14 Polar Function Graphs r = 2 (circle) rose curve r as a function of θ A polar graph plots r = f(θ): the radius depends on the angle. r = a (circle), r = a cos(nθ) (rose) Reading r = f(θ) As θ increases, r grows/shrinks → the curve spirals in/out. r < 0 points opposite θ. Common shapes: circles, roses, limaçons, spirals. A polar graph plots r = f(θ) — circles, roses, limaçons, and spirals. The Review Hub · AP Precalculus Unit 3 TOPIC 3.15 Rates of Change in Polar Functions How r changes with θ The average rate of change of r with respect to θ describes the spiral. Δr / Δθ over [θ₁, θ₂] r INCREASING → curve moves AWAY from the origin as θ grows. Interpreting the motion r DECREASING → the curve moves TOWARD the origin. Where r reaches a max/min, the distance from the origin is greatest/least. A negative r reflects the point across the origin. Estimate rate from a table or graph Compute Δr/Δθ between two angles to see how quickly the radius is changing over that interval. Δr/Δθ: r increasing → curve moves away from the origin; decreasing → toward it. The Review Hub · AP Precalculus Unit 3
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How to use the visual review

Spend 30 seconds per slide before clicking next. Look at the graph, then ask yourself: "Could I sketch this from memory and explain what its algebra is doing?"

Use the fullscreen button () on desktop for the best experience. Use arrow keys to navigate. Tap "Show all slides" to jump around.

This is great for review the night before the exam — fast, visual, and covers every graph feature you need to recognize in Unit 3.