g10-07-04 · Core topics
A square loses a sign
Does sin²θ+cos²θ=1 determine the sign of cosine?
1 MIN · PREDICT ON PAPER
Compare30° and150° before computing their squared coordinates.
Discuss or write first. No typed answer is required to continue.
The square gives the magnitude; the quadrant supplies the sign.
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 of squares equals1, but solving for cosine requires its quadrant sign. Opposite signs have the same square.
Conditions: The identity uses the same angle; square roots alone return a nonnegative value.
Mathematical justification: The unit-circle equation gives cos²θ+sin²θ=1. Thus |cosθ|=√(1−sin²θ), with the sign supplied by x-coordinate.
1 MIN · EXIT CHECK
If sinθ=1/2 and θ lies in quadrantII, find cosθ.
Check after answering
cosθ=−√3/2.
Teacher guidance and extension
Before: Angles, coordinates and basic function graphs.
Ask: Which extra fact distinguishes the two possible cosine values?
Watch for: Always using cosθ=√(1−sin²θ).
Challenge: Why does tanθ=sinθ/cosθ need an excluded case?
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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