LEARNER JOURNEY Example 1: cos225°
Locate225° is in Quadrant III.
Reference225°−180°=45°.
PointP=(−√2/2,−√2/2).
Read and checkcos225°=−√2/2; negative left of the y-axis.
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
Combine coterminal reduction, quadrant, reference angle, exact coordinates, and the x-coordinate definition.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- exact value
- A fraction or radical written without decimal rounding.
- normalize
- Replace a multi-turn angle with a convenient coterminal terminal measure.
- selected coordinate
- For cosine, the x-coordinate.
- sanity check
- Confirm sign and magnitude using position and the range [−1,1].
Look first, then name the mathematics
Build the visual meaning
- 225°
- Reference angle 45°
- Cosine reads x
- Point x = −√2/2, y = −√2/2
- cos 225° = −√2/2
Read the complete visual relationship as text
- The reference angle supplies magnitudes.
- The quadrant supplies signs.
- Cosine selects x.
- A sign and range check validates the answer.
Definition in plain language
For θ, determine its unit-circle point P=(x,y); then cosθ=x. Use special triangles for exact magnitudes and the quadrant for the sign.
Why this matters
A fixed process prevents coordinate swaps, sign errors, and premature calculator approximations.
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.
Reduce by full turns if necessary.
Identify quadrant or axis and reference angle.
Construct the exact point with correct signs.
Read x and verify sign/range.
The same method in words
- Reduce by full turns if necessary.
- Identify quadrant or axis and reference angle.
- Construct the exact point with correct signs.
- Read x and verify sign/range.
Interactive decision model
Build the reasoning, one decision at a time
Construct cos 5π/3.
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: cos120°
Locate120° is in Quadrant II.
Reference180°−120°=60°.
PointP=(−1/2,√3/2).
Read and checkcos120°=−1/2.
Example 3: cos(11π/6)
Recognize11π/6=330° in Quadrant IV.
Reference2π−11π/6=π/6.
PointP=(√3/2,−1/2).
Read and checkcos(11π/6)=√3/2, positive on the right.
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 coordinate does cosine read?
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 of cosine in Quadrant III, and why?
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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. Finish cos225° by reading x from (−√2/2,−√2/2).
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. Evaluate cos30°.
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Write a prediction before viewing the choices.
2. Evaluate cos60°.
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Write a prediction before viewing the choices.
3. Evaluate cos135°.
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Write a prediction before viewing the choices.
4. Evaluate cos210°.
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Write a prediction before viewing the choices.
5. Evaluate cos300°.
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Write a prediction before viewing the choices.
6. Evaluate cos(5π/4).
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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. Evaluate cos(7π/6).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Evaluate cos(2π/3).
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Write a prediction before viewing the choices.
3. Evaluate cos(3π/2).
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Write a prediction before viewing the choices.
4. Evaluate cos(17π/4).
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 cos120° not √3/2?
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
Evaluate cos(−17π/4) exactly and justify every step, including signed rotation and coterminal reduction.
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
Add 4π=16π/4 to −17π/4 to get −π/4, then add 2π to get 7π/4. This lies in Quadrant IV with reference angle π/4. Its point is (√2/2,−√2/2). Cosine reads x, so cos(−17π/4)=√2/2, positive on the right half.
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.