Unit 04 · Lesson 04 · Week 11 · Angles, Radians & the Unit Circle

Unit-Circle Coordinates and Reference Angles

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 unit-circle coordinates and reference angles. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intentions

Interpret the defining quantities and notation used in unit-circle coordinates and reference angles.
Construct or analyze at least two representations related to unit-circle coordinates and reference angles.
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

Exact circular valueFor θ=5π/6, the reference angle is π/6 and the point lies in Quadrant II, so cos θ=−√3/2 and sin θ=1/2.

Arc modelWith r=9 and θ=2π/3, arc length is s=rθ=6π and sector area is ½r²θ=27π.

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

Unit-Circle Coordinates and Reference Angles

Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.

U04 · L04

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.