LEARNER JOURNEY Example 1: find a positive coterminal angle for −40°
Choose operationAdd one full turn.
Calculate−40° + 360° = 320°.
Verify difference320° − (−40°) = 360°.
Conclude−40° and 320° are coterminal.
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
Identify angles that share a terminal side while keeping their different rotations clear.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- coterminal
- Having the same initial and terminal sides in standard position.
- full turn
- A rotation of 360° or 2π radians.
- family
- All angles θ + 360°k for integer k.
- integer
- A whole number, its negative, or zero.
Look first, then name the mathematics
Build the visual meaning
- 45°
- 405° = 360° + 45°
- 765° = 360° + 360° + 45°
- Same terminal side
- Full turn = 360°
Read the complete visual relationship as text
- Adding 360° adds one positive turn.
- Subtracting 360° adds one negative turn.
- The terminal side stays fixed.
Definition in plain language
Angles θ and θ + 360°k are coterminal for any integer k. In radians, the family is θ + 2πk.
Why this matters
Coterminal angles let you replace a difficult multi-turn measure with a familiar terminal measure without pretending the original rotation vanished.
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.
Add or subtract one or more complete turns.
Choose an integer k in θ + 360°k.
Verify the difference is an integer multiple of 360°.
State that the measures differ but their terminal sides match.
The same method in words
- Add or subtract one or more complete turns.
- Choose an integer k in θ + 360°k.
- Verify the difference is an integer multiple of 360°.
- State that the measures differ but their terminal sides match.
Interactive decision model
Build the reasoning, one decision at a time
Construct two coterminal angles for 75°.
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: reduce 1110°
Subtract turns1110° − 720° = 390°.
Continue390° − 360° = 30°.
Verify1110° − 30° = 1080° = 3 × 360°.
Conclude1110° is coterminal with 30°.
Example 3: test 50° and 770°
Find difference770° − 50° = 720°.
Recognize turns720° = 2 × 360°.
Apply definitionThe difference is a whole number of turns.
ConcludeThey are coterminal.
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 must coterminal angles share?
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2. What is one full turn in degrees?
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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. Finish: θ + ____ gives the degree family of coterminal angles.
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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. Give the least nonnegative angle coterminal with 410°.
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2. Give the least nonnegative angle coterminal with −25°.
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3. Which statement correctly compares 80° and 440°?
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4. Which statement correctly compares 80° and 260°?
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5. Give a negative angle coterminal with 120°.
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6. Reduce 1085° to [0°,360°).
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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 radian full-turn amount?
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2. Which statement correctly compares π/3 and 7π/3?
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3. Give a negative radian angle coterminal with π/6.
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4. How many full turns separate 30° and 1110°?
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5. Why are −30° and 330° not equal?
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Unfamiliar transfer
Explain, compare, and revise
A learner claims 1125° = 45° because both end at the same ray. Correct the claim using the coterminal formula and the rotation history.
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
1125° − 45° = 1080° = 360°(3). Therefore 1125° and 45° are coterminal. They are not equal measures: 1125° includes three full counterclockwise turns before the final 45°.
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.