g10-07-08 · Core topics
Double an angle, not its sine
How does sin2θ relate to sinθ?
1 MIN · PREDICT ON PAPER
At θ=45°, compare sin2θ,2sinθ and2sinθcosθ.
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
Putting α=β=θ into the sum formula gives sin2θ=2sinθcosθ and cos2θ=cos²θ−sin²θ.
Conditions: The identities hold for every real θ; 2sinθ alone is generally different.
Mathematical justification: Substitute equal angles into the proved angle-addition identities; no finite slider sweep is needed for the universal result.
1 MIN · EXIT CHECK
Find sin90° using θ=45° in the double-angle formula.
Check after answering
2(√2/2)(√2/2)=1.
Teacher guidance and extension
Before: Angles, coordinates and basic function graphs.
Ask: Which extra factor is lost in the incorrect formula?
Watch for: Doubling a function input by doubling its output.
Challenge: Derive cos2θ=1−2sin²θ.
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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