g10-07-10 · Core topics
Frequency squeezes the horizontal axis
Why is the period2π/|ω| rather than2πω?
1 MIN · PREDICT ON PAPER
Compare sinx,sin2x andsin(−2x) on the same interval.
Discuss or write first. No typed answer is required to continue.
The grey reference is sinx; compare one parameter at a time.
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 angle must change by2π to repeat. A nonzero ω therefore gives basic period2π/|ω|; negative ω reverses direction.
Conditions: This basic-period formula requires nonzero frequency and a nonzero sine coefficient. ω=0 gives a constant.
Mathematical justification: Solve |ω|T=2π for the least positive T. The sine function has least positive angular period2π.
1 MIN · EXIT CHECK
Find the basic period ofsin(3x).
Check after answering
2π/3.
Teacher guidance and extension
Before: Angles, coordinates and basic function graphs.
Ask: How much must x change to add one full turn inside the sine?
Watch for: Dividing by ω without an absolute value or dividing by0.
Challenge: Explain why a constant has periods but no least positive period.
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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