LEARN · EXPLAIN · REVISE

Classify the number of distinct solutions for every real k. Explain the exceptional role of k=0 in this closed interval.

Read the idea, work independently, then explain what changed.

高一必修第一册(A版).pdf · 5.4 · PDF 203 / printed page 196

Revisit first: Unit-circle coordinates and signs Identity choice and domain preservation Amplitude, period and phase in models

TOPIC 01

Complete trigonometric solution sets

Solve a transformed angle, recover every branch and intersect the original interval.

What you will be able to explain

Retain all angle branches and periodic copies in the requested interval.
Detect lost solutions from division and extraneous solutions from squaring.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

Examples 1–3: study a full demonstration; finish a solution from its first step; then choose a method independently before opening the solution. The remaining exercises may continue after class.

01 / Transfer#Your turn

Classify the number of distinct solutions for every real k. Explain the exceptional role of k=0 in this closed interval.

\sin x=k,\quad0\le x\le2\pi

Radians; interval closed at both ends.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Combine the sine range with endpoint inclusion.

Hint 2

Interior nonextreme heights cross twice; zero includes both endpoints.

Worked solution

Classify impossible and extremal levels.
$|k|>1:0;\quad |k|=1:1$
Treat ordinary levels and the duplicated endpoint height.
$0<|k|<1:2;\quad k=0:3$

Counts: 0,1,2,3 under the stated four cases.

Checks and common pitfalls: The endpoints are coterminal but distinct real inputs, so both count at zero.

Reasoning checklist · self / teacher assessment

Classify impossible and extremal levels.
Treat ordinary levels and the duplicated endpoint height.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Branch diagnosis	Use diagnostics to distinguish principal values and transformed intervals.
8	Build solution sets	Mark equal-height points and periodic copies before filtering.
10	Audit operations	Pairs contrast safe division in sin x=cos x with lost roots in sin 2x=cos x.
12	Classify and model	Groups classify parameter root counts and translate the wheel threshold.
5	Exit completeness	Require a complete set and a reason no other branch fits.

Question sequence

What interval does the transformed angle actually traverse?
Did this operation lose or add candidates?

Board development

Preserve factor branches.
$\cos x(2\sin x-1)=0$
Map an angle interval back to time.
$\pi/6<\pi t/4<5\pi/6$

Expected correct and incorrect responses

“Both endpoints count because they are different real x-values” correctly handles k=0.
“Divide by cosine immediately” risks losing valid zero-factor solutions.

Follow-up questions

Why does substituting all found roots not prove that none were missed?

Remediation

For missing branches, draw the horizontal value line before invoking inverse trigonometry.
For transformed-interval errors, put x bounds and angle bounds in separate columns.

Exit check

Solve cos 2x=1 for 0≤x≤2π. Expected 0,π,2π with closed endpoints.

Assessment checklist and sample responses

2 marks: branch and domain completeness.
Full: four roots for sin 2x=1/2 on [0°,360°); partial: first two only.
2 marks: validate nonreversible steps.
Full: reject positive-sine roots after squaring; partial: accept all four squared-equation roots.