LEARNER JOURNEY Example 1: signs at 140°
Locate140° is in Quadrant II.
HorizontalThe point is left, so x<0.
VerticalThe point is above, so y>0.
ConcludeThe sign pattern is (−,+).
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
Determine coordinate signs from left/right and above/below before attaching any exact magnitude.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- sign pattern
- The ordered signs (+,+), (−,+), (−,−), or (+,−). This tells how the angle or coordinate is interpreted in a solution.
- horizontal sign
- Positive to the right, negative to the left.
- vertical sign
- Positive above, negative below. This tells how the angle or coordinate is interpreted in a solution.
- axis point
- A point with one coordinate equal to 0.
Look first, then name the mathematics
Build the visual meaning
- Quadrant I: x +, y +
- Quadrant II: x −, y +
- Quadrant III: x −, y −
- Quadrant IV: x +, y −
- Right/left sets x; above/below sets y
Read the complete visual relationship as text
- Right of the y-axis means x>0.
- Left means x<0.
- Above the x-axis means y>0.
- Below means y<0.
Definition in plain language
Quadrant I has (+,+), II (−,+), III (−,−), and IV (+,−). Axis endpoints use 0 for the coordinate perpendicular to that axis.
Why this matters
Signs supply the direction information missing from special-angle magnitudes and determine the signs of sine and cosine.
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 point’s quadrant or axis.
Ask right or left to set x.
Ask above or below to set y.
Attach signs to known magnitudes and keep the order (x,y).
The same method in words
- Locate the point’s quadrant or axis.
- Ask right or left to set x.
- Ask above or below to set y.
- Attach signs to known magnitudes and keep the order (x,y).
Interactive decision model
Build the reasoning, one decision at a time
Construct the signed coordinates for a Quadrant II point with magnitudes √3/2 and 1/2.
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: signs at 250°
Locate250° is in Quadrant III.
HorizontalLeft gives x<0.
VerticalBelow gives y<0.
ConcludeThe sign pattern is (−,−).
Example 3: signs at 330°
Locate330° is in Quadrant IV.
HorizontalRight gives x>0.
VerticalBelow gives y<0.
ConcludeThe sign pattern is (+,−).
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 sign pattern in Quadrant I?
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 sign pattern in Quadrant II?
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 Quadrant III: x __ 0 and y __ 0.
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 sign pattern in Quadrant IV?
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Write a prediction before viewing the choices.
2. What is the sign of x at 120°, and why?
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3. What is the sign of y at 120°, and why?
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Write a prediction before viewing the choices.
4. At 225°, what signs do x and y have?
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5. At 300°, what signs do x and y have?
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6. At 90°, what is the sign/value pattern?
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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. At 180°, what is the sign/value pattern?
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Write a prediction before viewing the choices.
2. At −45°, what quadrant and pattern occur?
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3. At 585°, what sign pattern occurs?
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4. If a point is left and below, name its quadrant.
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Write a prediction before viewing the choices.
5. Why can an axis point not use a two-sign quadrant pattern?
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Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
A unit-circle point has a 60° reference angle and lies in Quadrant III. State its sign pattern, construct the exact point, and verify it.
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
Quadrant III is left and below, so the sign pattern is (−,−). A 60° reference angle has magnitudes (1/2,√3/2), giving P=(−1/2,−√3/2). Its squared coordinates add to 1/4+3/4=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.