g10-07-02 · Core topics
One circle, two coordinates
Which coordinate is sine, and which signs depend on the quadrant?
1 MIN · PREDICT ON PAPER
Compare sine and cosine at 30°,150°,210° and330°.
Discuss or write first. No typed answer is required to continue.
The point is (cosθ,sinθ).
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
The unit-circle point is (cosθ,sinθ). A full turn returns to the same point, while opposite points negate both coordinates.
Conditions: Angles may be negative or exceed360°; these are directed turns, not new radii.
Mathematical justification: Coordinates on x²+y²=1 have x=cosθ,y=sinθ by the unit-circle definition. Rotation by360° preserves the point.
1 MIN · EXIT CHECK
Give the coordinates at270°.
Check after answering
(0,−1), so cos270°=0 and sin270°=−1.
Teacher guidance and extension
Before: Angles, coordinates and basic function graphs.
Ask: Can a negative sine value still be a valid circle coordinate?
Watch for: Treating sine and cosine as positive lengths in every quadrant.
Challenge: Find all turns in[0°,720°] returning to(1,0).
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
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