LEARNER JOURNEY Example 1: convert 5π/6
Set up(5π/6)·(180°/π).
Cancelπ cancels.
Simplify180/6=30.
Conclude5·30°=150°.
Start here
Build the concept from the beginning
Starting point
No prior trigonometry is assumed. Every required term and decision is taught on this page.
Learning target
Reverse the conversion factor so radians cancel and the final measure is in degrees.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- reciprocal
- A fraction flipped so the numerator and denominator exchange places.
- approximate
- A rounded value, written with ≈.
- exact conversion
- A conversion completed without rounding.
- reasonableness check
- A comparison with landmarks such as π/2 and π.
Look first, then name the mathematics
Build the visual meaning
- 7π/12 radians
- 180°/π radians
- π and radians cancel
- 7 × 15°
- 105°
Read the complete visual relationship as text
- The factor equals 1.
- Radians cancel.
- A π factor often cancels exactly.
- Decimal radian inputs usually give approximate degrees.
Definition in plain language
For a radian measure r, degrees = r(180/π). Keep the sign, cancel radians and π, then simplify.
Why this matters
The units choose the reciprocal factor. Landmarks provide a strong check on any arithmetic result.
Reasoning path
See the method as a sequence
Move from left to right. Each decision prepares the next one; on a phone, follow the numbered cards from top to bottom.
Write the signed radian measure.
Multiply by 180°/(π radians).
Cancel radians and any π factor.
Simplify; use ≈ if a decimal or rounding is required.
The same method in words
- Write the signed radian measure.
- Multiply by 180°/(π radians).
- Cancel radians and any π factor.
- Simplify; use ≈ if a decimal or rounding is required.
Interactive decision model
Build the reasoning, one decision at a time
Construct the conversion of 11π/6 to degrees.
Choose this step, then check the construction.
Choose this step, then check the construction.
Choose this step, then check the construction.
The reasoning path changes as each decision is selected.
Choose every step, then check the construction.
Model before practice
Three fully explained examples
Each example names the decision, carries it out, and interprets the result.
Example 2: convert −7π/4
Preserve sign(−7π/4)·(180°/π).
Cancelπ cancels.
Simplify180/4=45.
Conclude−7·45°=−315°.
Example 3: convert 2.4 radians
Set up2.4·180°/π.
Recognize approximationThere is no symbolic π factor to cancel.
Calculate432°/π≈137.5°.
Check2.4 is between π/2 and π, so 137.5° lies between 90° and 180°.
Practice with decreasing support
Guided practice
Practice immediately after the model
Predict first. Then use the visual relationship and reasoned feedback to check two decisions.
1. Which factor converts radians to degrees?
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2. Convert π radians.
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Partially completed problem
Finish the missing step
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. Finish: π/2·180°/π = ____.
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Independent practice
Six problems for repetition
Solve all six without opening the worked examples unless feedback shows what to revise.
1. Convert π/6.
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2. Convert 2π/3.
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3. Convert 3π/4.
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4. Convert 7π/6.
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5. Convert 5π/3.
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6. Convert −π/4.
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Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. Convert 4π.
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2. Convert 13π/12.
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3. Convert 9π/2.
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4. Approximately convert 1 radian to the nearest tenth.
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5. Why must a decimal-radian conversion usually use ≈?
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Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Convert −25π/6 to degrees and explain both the signed rotation and a coterminal terminal angle.
Guided self-check
Judge the locked first attempt against every criterion. Each judgment gives revision guidance.
Choose Met or Needs revision.
Choose Met or Needs revision.
Choose Met or Needs revision.
Choose Met or Needs revision.
Model reasoning
(−25π/6)(180°/π)=−25·30°=−750°. This is a clockwise rotation. Adding 720° gives −30°, and adding another 360° gives the least nonnegative coterminal measure 330°. The measures are coterminal, not equal.
Write a first attempt before opening the self-check.
The record reveals exact answers and model reasoning only after the corresponding attempt or self-check has occurred.