Welch Teaching & Coaching

Trigonometry • College Preparatory Mathematics

Unit 3: Radian Measure

Define radians geometrically, rebuild the unit circle in radians, and use radians for exact values, arc length, and sector area.

U3

Unit Road Map

Lesson Map

Use this table as your route through the unit. Lesson numbers jump directly to the matching explanation.

Lesson	Topic	You should be able to…
3.1	What Is a Radian?	Connect arc length and radius to the definition of one radian.
3.2	Degrees ↔ Radians	Convert angle measures using 180° = π radians.
3.3	The Unit Circle in Radians	Label quadrantal and special angles in radians.
3.4	Reference Angles in Radians	Use quadrant structure to determine reference angles.
3.5	Exact & Approximate Values	Find exact special-angle values and calculator approximations for other angles.
3.6	Inverse Trig in Radians	Use calculators to determine angle measures when answers are requested in radians.
3.7	Arc Length	Use s=rθ when θ is measured in radians.
3.8	Sector Area	Use A=½r²θ when θ is measured in radians.

Navigation

How the Unit Is Built

Each resource below is part of the same learning cycle. Select a card to jump to it.

Lesson MapSee the sequence and jump directly to any lesson.Content SummaryRead the ideas, methods, examples, and checks.Video LibraryOpen a targeted Khan Academy or YouTube search for each lesson.VocabularyReview the language needed to read and explain the mathematics.Sample ProblemsPractice with expandable worked solutions.Equations & RulesUse a compact reference for formulas and decision rules.Review GuidePrepare with a skill checklist and fresh mixed practice.Performance TaskApply the unit in a modeling or measurement context.

Learn

Content Summary

Work through these lessons in order. Focus on the meaning of the mathematics, the decision that selects a method, and a check that confirms the result.

Lesson 3.1

What Is a Radian?

Connect arc length and radius to the definition of one radian.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 3.2

Degrees ↔ Radians

Convert angle measures using 180° = π radians.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 3.3

The Unit Circle in Radians

Label quadrantal and special angles in radians.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 3.4

Reference Angles in Radians

Use quadrant structure to determine reference angles.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 3.5

Exact & Approximate Values

Find exact special-angle values and calculator approximations for other angles.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 3.6

Inverse Trig in Radians

Use calculators to determine angle measures when answers are requested in radians.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 3.7

Arc Length

Use s=rθ when θ is measured in radians.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 3.8

Sector Area

Use A=½r²θ when θ is measured in radians.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Watch & Reinforce

Video Library

These links open focused video searches by lesson topic. Use them as reinforcement rather than as a substitute for showing your own reasoning.

Lesson 3.1

What Is a Radian?

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 3.2

Degrees ↔ Radians

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 3.3

The Unit Circle in Radians

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 3.4

Reference Angles in Radians

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 3.5

Exact & Approximate Values

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 3.6

Inverse Trig in Radians

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 3.7

Arc Length

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 3.8

Sector Area

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Language

Vocabulary

These terms should appear naturally in explanations, diagrams, and written reasoning.

radian
Write a definition in your own words and add an example or sketch in your STRIDES notes.

arc length
Write a definition in your own words and add an example or sketch in your STRIDES notes.

sector
Write a definition in your own words and add an example or sketch in your STRIDES notes.

central angle
Write a definition in your own words and add an example or sketch in your STRIDES notes.

reference angle
Write a definition in your own words and add an example or sketch in your STRIDES notes.

exact value
Write a definition in your own words and add an example or sketch in your STRIDES notes.

radian mode
Write a definition in your own words and add an example or sketch in your STRIDES notes.

Practice

Sample Problems

Try each problem before opening the solution. The examples are original practice aligned to the unit skills.

1

Convert to radians

Convert 225° to radians.

Show solution

225(π/180)=5π/4.

2

Convert to degrees

Convert 7π/6 to degrees.

Show solution

(7π/6)(180/π)=210°.

3

Reference angle

Find the reference angle for 5π/3.

Show solution

2π−5π/3=π/3.

4

Exact value

Find sin(7π/6).

Show solution

Reference angle π/6 and QIII sine is negative: −1/2.

5

Arc length

A circle has radius 9 cm and central angle 2π/3. Find arc length.

Show solution

s=9(2π/3)=6π cm.

6

Sector area

A sector has radius 8 and angle 3π/4. Find its area.

Show solution

A=½(64)(3π/4)=24π square units.

Reference

Equations & Rules

Know what each relationship means and when it applies; do not treat the box as a list of disconnected formulas.

Degree-Radian Equivalence

180° = π rad

Degrees → Radians

degrees × π/180

Radians → Degrees

radians × 180/π

Arc Length

s = rθ

Sector Area

A = ½r²θ

Full Circle

2π radians = 360°

Prepare

Review Guide

Skill Checklist

I can connect arc length and radius to the definition of one radian.
I can convert angle measures using 180° = π radians.
I can label quadrantal and special angles in radians.
I can use quadrant structure to determine reference angles.
I can find exact special-angle values and calculator approximations for other angles.
I can use calculators to determine angle measures when answers are requested in radians.
I can use s=rθ when θ is measured in radians.
I can use A=½r²θ when θ is measured in radians.

How to review

Complete the unit Skills Check packet without notes.
Mark the exact skill behind each error.
Return to that lesson and complete a fresh example.
Explain the correction in words, then retry a similar problem.
Finish with mixed practice so you must choose the method yourself.

Apply

Performance Task

Circular Motion Model

Model a rotating object such as a wheel, track curve, fan, or turntable. Connect angular measure in radians to distance traveled along an arc and area swept by a radius.

Evidence to include

A labeled mathematical model, organized calculations, appropriate units and precision, a written interpretation, and a check or discussion of limitations.