Tags: arc length, radian measure, evaluating trig without a calculator, special angles, sinusoidal model, period from context, midline, amplitude, rate of change, AP Precalculus Unit 3
Arc length on a circle equals the radius times the angle in radians. You can evaluate sine, cosine, and tangent at any multiple of π/6, π/4, or π/3 without a calculator by using reference angles and quadrant signs. Real-world periodic scenarios (jump ropes, sonar sweeps) are modelled with sinusoidal functions where you extract the period, midline, and amplitude from the context.
Arc length
The distance along the curved edge of a circle between two points. Calculated as s = rθ, where r is the radius and θ is the central angle in radians.
Radian
A unit of angle measure. One radian is the angle that subtends an arc equal in length to the radius. A full circle is 2π radians.
Reference angle
The positive acute angle between the terminal side of θ and the nearest part of the x-axis. Used to evaluate trig functions at any angle by applying the sign from the quadrant.
Special angles
The angles whose exact trig values you should know from memory: 0, π/6, π/4, π/3, π/2, and their equivalents in every quadrant.
Sinusoidal model
A function of the form y = A sin(B(x − C)) + D or y = A cos(B(x − C)) + D used to model real-world periodic behaviour (height, distance, temperature, etc.).
Rate of change (of a periodic function)
How quickly the function's output is changing at a given point. In concave-up regions the rate of change is increasing; in concave-down regions it is decreasing.
The arc length s between two points on a circle of radius r, separated by a central angle θ (in radians), is:
s = rθ
This is the formula you choose when asked "which equation could be used to find the arc length?" The answer is a = θr.
Example: a circle with radius 6 and a central angle of 7π/15. The arc from M to S travels from the positive x-axis (angle 0) up to angle 7π/15, but you need to determine exactly which arc is requested. If the angle from M to S going counterclockwise is (2π − 7π/15) = 23π/15, then:
s = 6 · (23π/15) = 138π/15 = 46π/5
Always check which arc is being asked for: the minor arc (shorter) or the major arc (longer), or the specific arc between two labelled points.
sin x = cos x when tan x = 1, which happens at x = π/4 + nπ for any integer n.
Among standard choices:
x = π/4 (Q I, both positive, tan = 1 ✓)
x = 5π/4 (Q III, both negative, tan = 1 ✓)
x = 9π/4 = 2π + π/4, which is coterminal with π/4 ✓
So if the options are 3π/4, 5π/4, 2π, and 9π/4, the answers are 5π/4 and 9π/4.
The process for every problem is the same three steps:
Step 1: Find the coterminal angle between 0 and 2π. Subtract or add multiples of 2π.
Step 2: Find the reference angle. This is the acute angle to the nearest x-axis.
Step 3: Apply the quadrant sign to the known value at the reference angle.
Reference values you need to know:
Angle | sin | cos | tan |
|---|---|---|---|
0 | 0 | 1 | 0 |
π/6 | 1/2 | √3/2 | √3/3 |
π/4 | √2/2 | √2/2 | 1 |
π/3 | √3/2 | 1/2 | √3 |
π/2 | 1 | 0 | undefined |
cos(2π/3): Quadrant II (reference angle π/3). Cosine is negative in Q II. cos(2π/3) = −cos(π/3) = −1/2
sin(11π/6): Quadrant IV (reference angle π/6). Sine is negative in Q IV. sin(11π/6) = −sin(π/6) = −1/2
sin(15π/2): Reduce: 15π/2 = 7(2π) + π/2. Coterminal with π/2. sin(π/2) = 1
cos(7π/6): Quadrant III (reference angle π/6). Cosine is negative in Q III. cos(7π/6) = −cos(π/6) = −√3/2
cos(−π/3): Coterminal with 2π − π/3 = 5π/3. Quadrant IV. Cosine is positive in Q IV. cos(−π/3) = cos(π/3) = 1/2
sin(3π/4): Quadrant II (reference angle π/4). Sine is positive in Q II. sin(3π/4) = sin(π/4) = √2/2
tan(5π/4): Quadrant III (reference angle π/4). Tangent is positive in Q III. tan(5π/4) = tan(π/4) = 1
tan(7π): Coterminal with π (since 7π = 3(2π) + π). tan(π) = 0
tan(4π/3): Quadrant III (reference angle π/3). Tangent is positive in Q III. tan(4π/3) = tan(π/3) = √3
When a word problem describes something that repeats (a jump rope, a sonar sweep, a Ferris wheel), extract these values from the context:
Maximum and minimum values: Read directly from the problem. For a jump rope starting on the ground (min = 0) and reaching 7.5 feet (max = 7.5).
Midline: y = (max + min) / 2 = (7.5 + 0) / 2 = 3.75 feet.
Amplitude: A = (max − min) / 2 = (7.5 − 0) / 2 = 3.75 feet.
Period: The time for one complete cycle. If 80 jumps happen in 60 seconds, each jump takes 60/80 = 0.75 seconds. The period is 0.75 seconds.
When the graph shows a periodic function with labelled points (F, G, J, K, P) and a midline:
The midline value equals the average distance (centre of sonar to edge of panel = 4 inches, so midline = 4).
The maximum distance is centre-to-edge + radius of sweep = 4 + 3 = 7 inches.
The minimum distance is centre-to-edge − radius = 4 − 3 = 1 inch.
5 rotations per second means the period = 1/5 = 0.2 seconds.
For the five points on such a graph:
F = top of the first cycle (a maximum): (t, 7)
G = midline, function decreasing: (t, 4)
J = bottom of the cycle (a minimum): (t, 1)
K = midline, function increasing: (t, 4)
P = top of the second cycle (next maximum): (t, 7)
Assign t-coordinates spaced at quarter-period intervals (0.05 seconds apart).
Between J (minimum) and K (midline crossing, function increasing):
The function is positive (all distances are positive in this context) and increasing.
The graph is concave down on this interval (it rises steeply at first, then levels off as it approaches the midline). So the rate of change is positive but decreasing.
Arc length: s = rθ (θ in radians)
Period from frequency: Period = 1 / frequency, or Period = total time / number of cycles
Midline: y = (max + min) / 2
Amplitude: A = (max − min) / 2
⚠️ The arc length formula only works when θ is in radians. If given degrees, convert first: θ_rad = θ_deg · π/180.
⚠️ When evaluating trig at angles like 15π/2, reduce by subtracting 2π repeatedly (or divide by 2π and take the remainder). Students often lose marks by skipping this step.
⚠️ cos(−θ) = cos(θ) (cosine is even), but sin(−θ) = −sin(θ) (sine is odd). This shortcut saves time on negative angles.
⚠️ In word problems, be careful about units. If the period is asked for, include units (seconds, minutes, etc.). If frequency is given as "80 per minute," convert to per-second or find the period in seconds.
⚠️ On the FRQ, "describe how the rate of change is changing" means describe the second derivative behaviour in words. Say whether the rate of change is increasing or decreasing, not just whether the function itself is increasing or decreasing.
⚠️ tan(π) = 0 and tan(0) = 0. Students sometimes confuse these with undefined values. Tangent is undefined at π/2 and 3π/2 (where cosine = 0).
Q: What is the arc length on a circle of radius 6 when the central angle is 7π/15?
A: s = rθ = 6(7π/15) = 42π/15 = 14π/5.
Q: Evaluate cos(2π/3) without a calculator.
A: Reference angle is π/3, quadrant II. cos(2π/3) = −1/2.
Q: Evaluate tan(5π/4) without a calculator.
A: Reference angle is π/4, quadrant III. tan(5π/4) = +1.
Q: For which value of x does sin x = cos x: 3π/4, 5π/4, 2π, or 9π/4?
A: sin x = cos x when tan x = 1. This occurs at 5π/4 and 9π/4. Among the options, both 5π/4 and 9π/4 work (though typically 9π/4 = π/4 + 2π is the expected "nice" answer in the first quadrant sense). If only one answer is accepted, 9π/4 is coterminal with π/4 where sin = cos = √2/2.
Q: Jessie completes 80 jumps in 60 seconds, with the rope reaching a max height of 7.5 feet from the ground. What is the period, and what is the midline?
A: Period = 60/80 = 0.75 seconds. Midline = (7.5 + 0)/2 = 3.75 feet.
Q: A sonar arm is 3 inches long, the screen centre is 4 inches from the panel edge, and it rotates 5 times per second. What are the maximum distance, minimum distance, and period of the function modelling distance to the panel edge?
A: Maximum = 4 + 3 = 7 inches. Minimum = 4 − 3 = 1 inch. Period = 1/5 = 0.2 seconds.
Q: On the interval from the minimum to the next midline crossing (function increasing), is the rate of change increasing or decreasing?
A: The function is concave down on this interval (it curves from a steep rise to a gentler slope approaching the midline), so the rate of change is decreasing.
arc length formula, s = rθ, radian, central angle, evaluating trig without calculator, reference angle, special angles, sin cos tan exact values, coterminal angle, sinusoidal model, periodic function application, jump rope problem, sonar problem, period, midline, amplitude, rate of change of periodic function, concavity, AP Precalculus, Unit 3, Topics 3.2, 3.3