LEARNER JOURNEY Example 1: point on positive x-axis
LocateThe point is one unit right of the origin.
Read xx=1.
Read yThere is no vertical displacement, so y=0.
ConcludeThe point is (1,0), and 1²+0²=1.
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
Read and construct points on the radius-1 circle using horizontal and vertical coordinates.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- unit circle
- The circle centered at (0,0) with radius 1.
- ordered pair
- A point written (x,y): horizontal coordinate first, vertical coordinate second.
- projection
- A perpendicular drop used to show one coordinate.
- terminal point
- The point where an angle’s terminal side meets the unit circle.
Look first, then name the mathematics
Build the visual meaning
- P = (cos θ, sin θ)
- sin θ = y
- cos θ = x
- radius = 1
Read the complete visual relationship as text
- Coordinates are ordered x first, y second.
- The radius from the origin has length 1.
- Horizontal and vertical projections form a right triangle.
Definition in plain language
For any terminal point P=(x,y) on the unit circle, x²+y²=1. The sign of x tells left or right; the sign of y tells below or above.
Why this matters
Coordinates turn angle position into exact numbers. Those numbers later become cosine and sine values.
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.
Locate the terminal point on the circle.
Read horizontal displacement as x.
Read vertical displacement as y.
Write (x,y) in order and verify x²+y²=1.
The same method in words
- Locate the terminal point on the circle.
- Read horizontal displacement as x.
- Read vertical displacement as y.
- Write (x,y) in order and verify x²+y²=1.
Interactive decision model
Build the reasoning, one decision at a time
Construct the point one unit above the origin.
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: point on negative y-axis
LocateThe point is one unit below the origin.
Read xThere is no horizontal displacement, so x=0.
Read yy=−1.
ConcludeThe point is (0,−1), and 0²+(−1)²=1.
Example 3: point in Quadrant II
Use signsLeft means x<0; above means y>0.
Use given magnitudesSuppose |x|=√3/2 and |y|=1/2.
Apply signsx=−√3/2 and y=1/2.
Verify3/4+1/4=1, so the point is on the unit circle.
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 radius of the unit circle?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Which coordinate is written first?
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 the unit-circle equation: x²+y²=____.
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 point is on the positive y-axis?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. What point is on the negative x-axis?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Which quadrant contains a point with x<0,y>0?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Which quadrant contains x>0,y<0?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Why does (3/5,4/5) lie on the unit circle?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
6. Why does (1,1) not lie on the unit circle?
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.
Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. If x=0 and the point is on the unit circle, what can y be?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. If y=0 on the unit circle, what can x be?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Give the sign pattern in Quadrant III.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Verify (√2/2,√2/2): what is x²+y²?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Why is (y,x) generally not the same point as (x,y)?
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.
Unfamiliar transfer
Explain, compare, and revise
A unit-circle point is left of the y-axis, above the x-axis, and has coordinate magnitudes √3/2 and 1/2 with the larger magnitude horizontal. Construct and verify the point.
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
Left and above means Quadrant II, so x is negative and y is positive. The larger horizontal magnitude is √3/2, so P=(−√3/2,1/2). Verification: (−√3/2)²+(1/2)²=3/4+1/4=1.
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.