LEARNER JOURNEY Example 1: verify (3/5,4/5)
Substitutex²+y²=(3/5)²+(4/5)².
Square9/25+16/25.
Add25/25=1.
ConcludeThe point lies on the unit circle.
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
Use right-triangle geometry to explain why every unit-circle point satisfies x²+y²=1.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- Pythagorean theorem
- For a right triangle with legs a,b and hypotenuse c, a²+b²=c².
- hypotenuse
- The side opposite the right angle; here, the radius.
- coordinate magnitude
- The nonnegative distance |x| or |y|.
- identity
- An equation true for every allowed value; here x²+y²=1 on the unit circle.
Look first, then name the mathematics
Build the visual meaning
- Hypotenuse r
- Vertical leg y
- Horizontal leg x
- x² + y² = r²
- When r = 1: x² + y² = 1
Read the complete visual relationship as text
- The horizontal leg has length |x|.
- The vertical leg has length |y|.
- The radius is the hypotenuse r.
- Squaring removes coordinate signs.
Definition in plain language
Dropping perpendiculars from P=(x,y) makes a right triangle with leg lengths |x| and |y| and hypotenuse 1. Thus |x|²+|y|²=1², which is x²+y²=1.
Why this matters
This relationship verifies coordinates and lets you find a missing coordinate when the quadrant supplies its sign.
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 x²+y²=1.
Substitute the known coordinate.
Isolate the square of the unknown coordinate.
Take both square-root possibilities, then use the quadrant to choose the sign.
The same method in words
- Write x²+y²=1.
- Substitute the known coordinate.
- Isolate the square of the unknown coordinate.
- Take both square-root possibilities, then use the quadrant to choose the sign.
Interactive decision model
Build the reasoning, one decision at a time
Construct a Quadrant IV point with x=4/5.
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: find y when x=3/5 in Quadrant I
Equation(3/5)²+y²=1.
Isolatey²=1−9/25=16/25.
Square rooty=±4/5.
Use quadrantQuadrant I has y>0, so y=4/5.
Example 3: find x when y=−1/2 in Quadrant III
Equationx²+(−1/2)²=1.
Isolatex²=1−1/4=3/4.
Square rootx=±√3/2.
Use quadrantQuadrant III has x<0, so x=−√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. Which side of the coordinate right triangle is the radius?
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 holds on the unit circle?
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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: if x=0, then y²=____.
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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. Why does (5/13,12/13) satisfy x²+y²=1?
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2. Why does (1/2,1/2) not lie on the unit circle?
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3. If x=√3/2, what is |y|?
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4. If y=√2/2, what is |x|?
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5. If x=−8/17, what is |y|?
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6. What sign must y have in Quadrant II, and why?
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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 sign must x have in Quadrant III, and why?
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Write a prediction before viewing the choices.
2. Find y for x=0.6 in Quadrant IV.
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3. Find x for y=−5/13 in Quadrant III.
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4. Why do negative coordinates become positive when squared?
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5. Why is x+y=1 not the unit-circle equation?
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Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
A point lies on the unit circle in Quadrant II and has y=12/13. Find x and justify its sign.
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
x²+(12/13)²=1, so x²=1−144/169=25/169. Thus x=±5/13 algebraically. Quadrant II is left of the y-axis, so x must be negative. The point is (−5/13,12/13).
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.