LEARNER JOURNEY Example 1: cos 60°
PointP60=(1/2,√3/2).
SelectCosine reads x, the first coordinate.
Resultcos60°=1/2.
Check60° is on the right half, so cosine is positive.
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
Understand cosine as horizontal position on the unit circle, including its sign and range.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- cosine
- The x-coordinate of the unit-circle terminal point.
- horizontal projection
- The left-right component from the origin to the point.
- range
- All possible cosine values, from −1 through 1.
- zero of cosine
- An angle whose terminal point lies on the y-axis.
Look first, then name the mathematics
Build the visual meaning
- P = (x, y)
- radius = 1
- Horizontal coordinate
- cos θ = x
Read the complete visual relationship as text
- Cosine reads the first coordinate.
- Right-half points have positive cosine.
- Left-half points have negative cosine.
- Top and bottom axis points have cosine 0.
Definition in plain language
Follow the terminal side to P=(x,y). Read the first coordinate: cos θ=x. The radius is 1, so −1≤cos θ≤1.
Why this matters
This definition explains signs, exact values, and why cosine tracks horizontal rather than vertical movement.
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.
Find the terminal point or its quadrant/reference angle.
Construct or read the ordered pair (x,y).
Select the first coordinate x.
Check its sign from left/right and its magnitude against the interval [−1,1].
The same method in words
- Find the terminal point or its quadrant/reference angle.
- Construct or read the ordered pair (x,y).
- Select the first coordinate x.
- Check its sign from left/right and its magnitude against the interval [−1,1].
Interactive decision model
Build the reasoning, one decision at a time
Construct cos 225° from the unit-circle point.
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: cos 150°
Reference150° has reference 30°.
PointP=(−√3/2,1/2).
SelectRead x.
Resultcos150°=−√3/2, negative on the left half.
Example 3: cos 270°
Locate270° ends at (0,−1).
SelectThe x-coordinate is 0.
Resultcos270°=0.
InterpretThe bottom point has no horizontal displacement.
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 coordinate defines cosine?
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2. What is the range of cosine?
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Partially completed problem
Finish the missing step
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. Complete: if P=(−3/5,4/5), then cosθ=____.
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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. Find cos0°.
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2. Find cos90°.
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3. Find cos180°.
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4. Find cos45°.
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5. Find cos120°.
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6. Find cos300°.
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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 sign of cosine in Quadrant IV, and why?
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2. Find cos225°.
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3. Find cos(3π/2).
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4. If cosθ=0, where is the terminal point?
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5. Why can cosθ never equal 2 on the unit circle?
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Unfamiliar transfer
Explain, compare, and revise
Evaluate cos(7π/6) from the definition, explaining the angle, point, selected coordinate, and sign check.
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
7π/6=210°, which lies in Quadrant III with reference angle 30°. The exact point is (−√3/2,−1/2). Cosine is the x-coordinate, so cos(7π/6)=−√3/2. The negative sign is correct because the terminal point lies left of the y-axis.
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.