LEARNER JOURNEY Example 1: derive the Quadrant I point
Equal legsSet x=y=a.
Pythagorean equationa²+a²=1.
Solve2a²=1, so a=√2/2.
ConcludeP=(√2/2,√2/2).
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
Derive the 45° unit-circle point from an isosceles right triangle instead of memorizing it.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- isosceles right triangle
- A right triangle with two equal legs and two 45° angles.
- equal-leg variable
- A single variable used for both equal legs.
- principal square root
- The nonnegative square root used for a Quadrant I length.
- rationalized form
- √2/2, equivalent to 1/√2.
Look first, then name the mathematics
Build the visual meaning
- radius = 1
- 45°
- x = y = √2/2
Read the complete visual relationship as text
- Both acute triangle angles are 45°.
- Opposite equal angles have equal legs.
- The unit radius is the hypotenuse.
Definition in plain language
Let each leg be a. Then a²+a²=1, so 2a²=1, a²=1/2, and a=√(1/2)=√2/2. Both coordinates are positive in Quadrant I.
Why this matters
This derivation provides the magnitude used at every 45° reference angle, with quadrant signs applied afterward.
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.
Recognize a 45-45-90 triangle.
Set x=y=a.
Solve 2a²=1 for the positive length a.
Write the ordered pair and apply new signs only when moving to another quadrant.
The same method in words
- Recognize a 45-45-90 triangle.
- Set x=y=a.
- Solve 2a²=1 for the positive length a.
- Write the ordered pair and apply new signs only when moving to another quadrant.
Interactive decision model
Build the reasoning, one decision at a time
Construct the 225° unit-circle point from the 45° triangle.
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: build 135°
Reference angle135° has reference angle 45°.
MagnitudesBoth coordinate magnitudes are √2/2.
SignsQuadrant II has x<0,y>0.
ConcludeP=(−√2/2,√2/2).
Example 3: build 315°
Reference angle360°−315°=45°.
MagnitudesBoth magnitudes are √2/2.
SignsQuadrant IV has x>0,y<0.
ConcludeP=(√2/2,−√2/2).
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. Why are the legs equal in the 45° triangle?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. What equation results after setting x=y=a?
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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: a²=1/2, so a=____.
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. What is the 45° unit-circle point?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. What is the 135° point?
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Write a prediction before viewing the choices.
3. What is the 225° point?
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Write a prediction before viewing the choices.
4. What is the 315° point?
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Write a prediction before viewing the choices.
5. Verify the 45° point: what is (√2/2)²+(√2/2)²?
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Write a prediction before viewing the choices.
6. What is the reference angle of 675°?
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Not completed yet.
Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. What coordinate magnitudes occur for any 45° reference angle?
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Write a prediction before viewing the choices.
2. Why is (1/2,1/2) not the 45° point?
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Write a prediction before viewing the choices.
3. At 45°, which coordinate is larger?
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Write a prediction before viewing the choices.
4. Give the 45° magnitudes in decimal form to three decimals.
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5. Why does the derivation use the positive root in Quadrant I?
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Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Derive the exact unit-circle point for 585° without quoting a memorized coordinate.
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
585°−360°=225°, which lies in Quadrant III and has reference angle 45°. A 45-45-90 triangle has equal legs a, so a²+a²=1, 2a²=1, and a=√2/2. Quadrant III makes both coordinates negative: P=(−√2/2,−√2/2).
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.