LEARNER JOURNEY Example 1: derive the 30° point
Scale1:√3:2 becomes 1/2:√3/2:1.
Opposite sideThe side opposite 30° is the vertical short leg 1/2.
Adjacent sideThe horizontal long leg is √3/2.
ConcludeP30=(√3/2,1/2).
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
Scale a 30-60-90 triangle to a unit hypotenuse and connect its leg lengths to coordinates.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- 30-60-90 triangle
- A right triangle with acute angles 30° and 60°.
- side ratio
- short leg : long leg : hypotenuse = 1:√3:2.
- scale factor
- The multiplier applied to every side; here 1/2.
- coordinate order
- At 30°, horizontal is long and vertical is short; at 60°, they swap.
Look first, then name the mathematics
Build the visual meaning
- radius = 1
- 30°
- 60°
- √3/2
- 1/2
- (√3/2, 1/2)
Read the complete visual relationship as text
- The short leg is opposite 30°.
- The long leg is opposite 60°.
- The hypotenuse is 2 in the base ratio and 1 on the unit circle.
Definition in plain language
Divide every side in 1:√3:2 by 2 to make the hypotenuse 1. This gives short leg 1/2 and long leg √3/2.
Why this matters
The derivation explains both 30° and 60° coordinate magnitudes and prevents swapping them.
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 base ratio 1:√3:2.
Scale by 1/2 so the hypotenuse becomes 1.
Place the side opposite the named angle.
Read horizontal x and vertical y, then apply quadrant signs if needed.
The same method in words
- Write the base ratio 1:√3:2.
- Scale by 1/2 so the hypotenuse becomes 1.
- Place the side opposite the named angle.
- Read horizontal x and vertical y, then apply quadrant signs if needed.
Interactive decision model
Build the reasoning, one decision at a time
Construct the point at 330°.
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 the 60° point
Use same sidesThe magnitudes remain 1/2 and √3/2.
Opposite sideThe side opposite 60° is the vertical long leg √3/2.
Adjacent sideThe horizontal short leg is 1/2.
ConcludeP60=(1/2,√3/2).
Example 3: build 240°
Reference angle240°−180°=60°.
60° magnitudes(1/2,√3/2).
Quadrant signsQuadrant III makes both negative.
ConcludeP240=(−1/2,−√3/2).
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 is the base 30-60-90 side ratio?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. What scale factor makes the hypotenuse 1?
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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. Finish: the scaled long leg is ____.
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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. What is the 30° point?
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Write a prediction before viewing the choices.
2. What is the 60° point?
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3. How do the coordinate magnitudes compare at 30°?
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4. How do the coordinate magnitudes compare at 60°?
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5. What is the 120° point?
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6. What is the 150° point?
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Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. What is the 210° point?
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Write a prediction before viewing the choices.
2. What is the 300° point?
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3. Verify the 30° point: what is 3/4+1/4?
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4. Which side lies opposite 30°?
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5. Why do 30° and 60° swap coordinate magnitudes?
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Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Derive and construct the exact unit-circle point for 300° from the 30-60-90 ratio.
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
Scaling 1:√3:2 by 1/2 gives 1/2:√3/2:1. Since 300° has reference angle 60°, the Quadrant I magnitudes would be (1/2,√3/2). At 300° the point is in Quadrant IV, so x is positive and y negative: P=(1/2,−√3/2).
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.