g10-07-03 · Core topics
Reflect an angle without recalculating
Why do θ and180°−θ share a sine but not always a cosine?
1 MIN · PREDICT ON PAPER
Predict which coordinate changes under reflection in the y-axis.
Discuss or write first. No typed answer is required to continue.
Reflection keeps y and reverses x.
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
Reflection in the y-axis maps(cosθ,sinθ) to(−cosθ,sinθ), giving the supplementary-angle formulas.
Conditions: The identities hold for every real angle; matching plotted decimals is only a check.
Mathematical justification: The reflected ray makes angle π−θ and keeps y while reversing x. Therefore sin(π−θ)=sinθ and cos(π−θ)=−cosθ.
1 MIN · EXIT CHECK
Express cos150° in terms of cos30°.
Check after answering
cos150°=−cos30°=−√3/2.
Teacher guidance and extension
Before: Angles, coordinates and basic function graphs.
Ask: What happens when the point is already on the y-axis?
Watch for: Assuming equal sines imply equal angles.
Challenge: Explain sin(θ+180°)=−sinθ using a half-turn.
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.
Optional local notebook
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