Use a reference angle and quadrant for a full turn
Read sine and cosine as unit-circle coordinates · Choose signs using the quadrant · Find every allowed angle in a stated interval
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Exact values
- Cartesian coordinates and rotations
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Read sine and cosine as unit-circle coordinates
- Choose signs using the quadrant
- Find every allowed angle in a stated interval
Coordinates set the signs
On the unit circle, cosine is the horizontal coordinate and sine the vertical coordinate. Sine is positive in the upper half-plane and negative in the lower; cosine is positive on the right and negative on the left. For 240°, the reference angle is 60° and the point is in quadrant three, so sin240°=−√3/2 and cos240°=−1/2.
One value can have several angles
A calculator inverse returns a principal value, not every angle in a full turn. For sinθ=1/2 on 0°≤θ≤360°, both 30° and 150° satisfy the equation. For cosθ=−√2/2, both 135° and 225° satisfy it. Follow the stated interval: on this closed interval, cosθ=1 includes 0° and 360°, while a half-open interval 0°≤θ<360° includes only 0°.
Sources for this lesson
Pause and explain
Can sinθ=1/2 have two angles in a full turn?
Worked example
Find exact sin240° and cos240°.
Show the worked solution
- Use reference angle 60°.
- Both coordinates are negative in quadrant three.
Answer −√3/2; −1/2.
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find exact sin210° and cos210°.
Give me a hint
Use reference angle 30° and quadrant three.
Compare my reasoning
- Sine is −1/2.
- Cosine is −√3/2.
sin210°=−1/2; cos210°=−√3/2.
Look for these in your work
- Identify the quadrant
- Apply signs to exact values
Solve cosθ=−√2/2 on 0°≤θ≤360°.
Give me a hint
Cosine is negative on the left, with reference angle 45°.
Compare my reasoning
- Quadrant two gives 180°−45°=135°.
- Quadrant three gives 180°+45°=225°.
135° and 225°.
Look for these in your work
- Find all allowed angles
- Verify their signs
Solve sinθ=0 and cosθ=1 on 0°≤θ≤360°. Explain how excluding 360° changes the lists.
Give me a hint
Use the circle axis crossings and the interval endpoints.
Compare my reasoning
- Sine zero occurs at 0°,180°,360°; cosine one at 0°,360°.
- If θ<360° is required, remove 360° from each list.
sinθ=0: 0°,180°,360°; cosθ=1: 0°,360°; omit 360° for the half-open interval.
Look for these in your work
- Handle endpoint inclusion
- Distinguish the two equations
Common mistakes
- An acute reference angle alone does not fix the sign.
- Check interval endpoints and all possible quadrants.
What can you explain now?
Can sinθ=1/2 have two angles in a full turn?
Compare with the explanation
Yes; 30° and 150° have the same sine.
Their unit-circle points have the same vertical coordinate.
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Make it stick
Can sinθ=1/2 have two angles in a full turn?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.