g10-07-07 · Core topics
Can sine distribute over addition?
Is sin(α+β)=sinα+sinβ?
1 MIN · PREDICT ON PAPER
Test α=β=30° mentally before revealing the result.
Discuss or write first. No typed answer is required to continue.
The product identity agrees; the simple additive claim generally fails.
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 sum-angle identity is sin(α+β)=sinαcosβ+cosαsinβ. Sine is not an additive function.
Conditions: All angles must use the same unit. A successful special case such as β=0 is not a general proof.
Mathematical justification: Rotate(cosα,sinα) by β. Its new y-coordinate is sinαcosβ+cosαsinβ, which is sin(α+β).
1 MIN · EXIT CHECK
Give an exact counterexample to the additive claim.
Check after answering
α=β=30°: sin60°=√3/2, but sin30°+sin30°=1.
Teacher guidance and extension
Before: Angles, coordinates and basic function graphs.
Ask: Which products account for the rotation of each original coordinate?
Watch for: Replacing sin(α+β) with sinα+sinβ.
Challenge: Use β=−α to obtain a difference-angle relation.
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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