LEARNER JOURNEY Example 1: derive 90°
Anchor180° = π radians.
DivideDivide both sides by 2.
Simplify180°/2=90° and π/2 remains exact.
Conclude90° = π/2 radians.
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
Build a small set of equivalent degree-radian landmarks from one half-turn relationship.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- landmark
- A frequently used angle whose degree and radian measures are known exactly.
- π
- The circle constant, approximately 3.14159.
- equivalent measures
- Different units describing the same rotation.
- exact form
- A value written with π rather than a rounded decimal.
Look first, then name the mathematics
Build the visual meaning
- 0° = 0
- 90° = π/2
- 180° = π
- 270° = 3π/2
- 360° = 2π
Read the complete visual relationship as text
- A full turn is 360° = 2π.
- A half turn is 180° = π.
- A quarter turn is 90° = π/2.
- Common special angles divide the half turn into equal pieces.
Definition in plain language
Since 180° and π radians describe the same half turn, divide or multiply both sides together to derive other pairs.
Why this matters
Landmarks make conversion, quadrant recognition, and exact sine/cosine evaluation faster while remaining grounded in the half-turn definition.
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.
Start with 180° = π.
Divide both sides by the same number to get a smaller landmark.
Multiply a known landmark to get larger angles.
Check the location: the degree and radian forms must end on the same ray.
The same method in words
- Start with 180° = π.
- Divide both sides by the same number to get a smaller landmark.
- Multiply a known landmark to get larger angles.
- Check the location: the degree and radian forms must end on the same ray.
Interactive decision model
Build the reasoning, one decision at a time
Construct the landmark for 225°.
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: derive 30°
Anchor180° = π.
DivideDivide both sides by 6.
Simplify180°/6=30°.
Conclude30° = π/6.
Example 3: derive 270°
Use 90°90° = π/2.
MultiplyMultiply both sides by 3.
Simplify3·90°=270° and 3·π/2=3π/2.
Conclude270° = 3π/2.
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. What radian measure equals 180°?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. What radian measure equals 360°?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Partially completed problem
Finish the missing step
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. Complete: 90° = ____.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Independent practice
Six problems for repetition
Solve all six without opening the worked examples unless feedback shows what to revise.
1. Convert 45° using a landmark.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Convert 60° using a landmark.
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Write a prediction before viewing the choices.
3. Convert 120° using landmarks.
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Write a prediction before viewing the choices.
4. Convert 150° using landmarks.
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Write a prediction before viewing the choices.
5. Convert 210° using landmarks.
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Write a prediction before viewing the choices.
6. Convert 300° using landmarks.
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Write a prediction before viewing the choices.
Not completed yet.
Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. Convert 315° using landmarks.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. What degree measure equals 3π/2?
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Write a prediction before viewing the choices.
3. What degree measure equals 5π/6?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. What degree measure equals 7π/4?
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Write a prediction before viewing the choices.
5. Why is π/3 = 60° exact rather than approximate?
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Write a prediction before viewing the choices.
Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Derive the degree-radian pairs for 30°, 150°, and 330° from 180° = π without using a decimal approximation.
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
Divide 180°=π radians by 6 to get 30°=π/6. Then 150°=5(30°)=5π/6 radians, and 330°=11(30°)=11π/6 radians. Every step uses exact multiples of the same landmark.
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.