Lesson 17: Periodic Phenomena, Radians & the Unit Circle

Warm-Up

Things that repeat: your heartbeat (~0.8 s per beat), a Ferris wheel (one lap every 40 s), daylight hours (one cycle per 365 days), an A-note's air pressure (1/440 s per vibration).

Wildly different time scales — identical mathematics. Each has a period: the length of one complete cycle, after which everything replays exactly. Unit 3 is the study of functions with this replay property, and it opens by rebuilding the machinery — angles, radians, and a circle of radius one — that makes periodic functions computable.

Core Concept

Periodic functions

f is periodic with period T if f(x + T) = f(x) for all x — the graph repeats every T units, and T is the smallest such positive number. From any graph or context: the period is the horizontal length of one full cycle (peak to peak, or any point to its next exact repeat). Frequency = 1/T, cycles per unit input. A wheel making 3 revolutions per minute has period 60/3 = 20 seconds.

Radians: the natural angle unit

Place an angle at the center of a circle of radius r. The radian measure of the angle is

θ = arc length / radius = s/r

— how many radius-lengths of arc the angle cuts. A full circle: circumference/radius = 2πr/r = 2π radians = 360°. So:

180° = π rad        degrees → radians: multiply by π/180        radians → degrees: multiply by 180/π

The special angles, both languages (memorize cold):

Arc length, rearranged: s = rθ (θ in radians) — a formula the exam uses directly.

The unit circle: sine and cosine as coordinates

Put an angle θ in standard position (vertex at origin, initial ray along positive x-axis, counterclockwise = positive). Where its terminal ray crosses the unit circle (radius 1):

that point is  (cos θ, sin θ)

cos θ = x-coordinate; sin θ = y-coordinate. Every trig fact for the next eight lessons flows from this definition:

• cos²θ + sin²θ = 1 (the point is on the circle: x² + y² = 1)
• Signs by quadrant: QI (+,+), QII (−,+), QIII (−,−), QIV (+,−). Mnemonic "All Students Take Calculus": All, Sin, Tan, Cos positive in QI–QIV respectively.
• Periodicity built in: θ and θ + 2π land on the same point → sin and cos have period 2π
• cos(−θ) = cos θ (even), sin(−θ) = −sin θ (odd): reflecting θ across the x-axis keeps x, flips y

Tangent: tan θ = sin θ/cos θ = y/x = the slope of the terminal ray. Undefined where cos θ = 0 (vertical ray: θ = π/2, 3π/2, …); period π (a ray and its opposite have the same slope).

Special-angle values

First-quadrant column (then reflect for the rest):

Pattern: sin runs √0/2, √1/2, √2/2, √3/2, √4/2; cos runs the same list backwards.

Reference angles extend the table to all quadrants: for any θ, find the acute angle between the terminal ray and the x-axis; take the table value; apply the quadrant's sign. Example: 7π/6 is π/6 past π (QIII, both coordinates negative): sin(7π/6) = −1/2, cos(7π/6) = −√3/2.

[GRAPH: Unit circle with the 16 special points marked. QI points labeled with coordinates: (√3/2, 1/2) at π/6, (√2/2, √2/2) at π/4, (1/2, √3/2) at π/3. The angle 7π/6 drawn with terminal ray in QIII, reference angle π/6 shaded, terminal point (−√3/2, −1/2) labeled. Axes points (1,0), (0,1), (−1,0), (0,−1) at 0, π/2, π, 3π/2.]

Worked Examples

Example 1 (easy) — Conversions 🚫 No-Calc

Problem: Convert 150° to radians and 3π/4 to degrees.

Solution: 150·(π/180) = 5π/6. And (3π/4)(180/π) = 3·45 = 135°.

Interpretation: Reduce before multiplying: 150/180 = 5/6, done. Fluency here is speed everywhere in Unit 3.

Example 2 (medium) — Full quadrant tour of one reference angle 🚫 No-Calc

Problem: Evaluate sin θ and cos θ at all four angles with reference angle π/3: θ = π/3, 2π/3, 4π/3, and 5π/3.

Solution: Reference values: sin = √3/2, cos = 1/2. Apply quadrant signs:

Interpretation: One memorized pair + four sign patterns = eight values. This is the entire quadrant system in one table.

Example 3 (medium) — Arc length in context 🚫 No-Calc

Problem: A pizza cutter wheel of radius 6 cm rolls through an angle of 2π/3. How far does a point on its rim travel along the arc?

Solution: s = rθ = 6·(2π/3) = 4π cm ≈ 12.57 cm.

Interpretation: θ must be in radians for s = rθ — that's what radians are for. (In degrees you'd need the clunky s = (θ/360)·2πr.)

Example 4 (AP-style) — Tangent as slope 🚫 No-Calc

Problem: The terminal ray of angle θ passes through the unit-circle point (−√2/2, √2/2). Find θ (smallest positive), tan θ, and interpret tan θ geometrically.

Solution: x < 0, y > 0: QII with reference angle π/4 → θ = 3π/4. tan θ = y/x = (√2/2)/(−√2/2) = −1: the terminal ray has slope −1 (it's the line y = −x, upper-left half).

Interpretation: Tangent questions become slope questions: rise (sin) over run (cos). The unit circle is a picture of all three functions at once.

Common Mistakes

1. Degree mode on a radian exam. sin(π/6) in degree mode returns ≈ 0.00914, not 0.5 — garbage that looks computed. Fix: set radian mode now, check it at the start of Part B, every time. AP Precalc defaults to radians.
2. sin and cos coordinates swapped. cos is x (first coordinate), sin is y (second). Anchor: "cos comes first alphabetically and in the coordinate pair."
3. Sign errors outside QI. Values are easy; signs kill. Fix: sketch the terminal ray, note the quadrant's (±, ±) before writing anything.
4. Reference angle measured from the y-axis. Reference angles are always measured to the x-axis. 5π/6 has reference π/6 (distance to π), not π/3.
5. tan period 2π. Tangent repeats every π — opposite rays share a slope. Its values also aren't bounded by ±1 (that's sin/cos only).

Practice Problems

12. (FRQ-style) 🚫 The angle θ = 5π/4 is in standard position. (i) State the quadrant of θ's terminal ray and the reference angle. (ii) Find the coordinates of the point where the terminal ray meets the unit circle, and the values of sin θ, cos θ, and tan θ. (iii) Find one positive and one negative angle coterminal with θ, and explain why all coterminal angles have identical sine and cosine values.

FRQ Practice — Task Model: Communicating about Functions (FRQ 4 style) 🚫 No-Calc

Let P(θ) = (cos θ, sin θ) denote the point on the unit circle for angle θ in standard position.

(a) (i) For θ in the open interval (π/2, π), state the signs of cos θ and sin θ. (ii) Explain each sign using the location of P(θ).

(b) As θ increases from π/2 to π: (i) does sin θ increase or decrease? (ii) Justify using the motion of the point P(θ) along the circle.

(c) (i) Explain, using the unit circle, why sin(θ + 2π) = sin θ for every θ. (ii) A student claims tan(θ + 2π) = tan θ proves that the period of tangent is 2π. Explain the flaw.

Model Response & Rubric (6 points)

(a) [2 pts] (i) [1 pt] cos θ < 0 and sin θ > 0. (ii) [1 pt] For θ in (π/2, π), P(θ) lies in the second quadrant: to the left of the y-axis (x = cos θ negative) and above the x-axis (y = sin θ positive).

(b) [2 pts] (i) [1 pt] sin θ decreases (from 1 toward 0). (ii) [1 pt] As θ goes from π/2 to π, P(θ) travels from (0, 1) counterclockwise to (−1, 0); its height — the y-coordinate, which is sin θ — falls from 1 to 0 throughout.

(c) [2 pts] (i) [1 pt] Increasing θ by 2π carries P once completely around the circle, returning it to the identical point; identical point → identical y-coordinate → sin(θ + 2π) = sin θ. (ii) [1 pt] The period is the smallest positive shift that repeats the function. tan repeats every 2π, but it also repeats every π (θ and θ + π give opposite rays with the same slope: tan(θ + π) = (−sin θ)/(−cos θ) = tan θ), so the period is π. Showing that 2π works only shows 2π is a repeat, not the smallest one.