Unit 04 · Lesson 03 · Week 10 · Angles, Radians & the Unit Circle
Arc Length, Sector Area & Angular Speed
Measure rotation in radians and use the unit circle to connect angle, coordinate, arc length, angular velocity, and exact trigonometric values. In this lesson, the central focus is arc length, sector area & angular speed. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.
Learning intentions
- Interpret the defining quantities and notation used in arc length, sector area & angular speed.
- Construct or analyze at least two representations related to arc length, sector area & angular speed.
- Show symbolic work with correct domains, units, intervals, or angle measures.
- Explain why the selected method is valid and check the result against another representation.
Essential question
How does circular measure connect angles, coordinates, arc length, angular velocity, and exact trigonometric values?
Multiple-representation model
Measure rotation in radians and use the unit circle to connect angle, coordinate, arc length, angular velocity, and exact trigonometric values.
s=r\thetaA=\frac12r^2\theta(x,y)=(\cos\theta,\sin\theta)\omega=\frac{\theta}{t}
Worked reasoning
Misconception check: Radians are dimensionless ratios of arc length to radius; they are not written with a degree symbol.
Communication and reasoning
Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.
Khaemenes Academy · Open-Age Precalculus
Arc Length, Sector Area & Angular Speed
Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.
Name
Date
Convert 150° to radians.
Convert 7π/4 to degrees.
Find an angle coterminal with −π/3 in [0,2π).
Find sin(5π/6).
Find cos(7π/4).
A wheel of radius 4 turns through 3π/2 radians. Find arc length.