Learning goal and prerequisites
Find all branches and respect included and excluded interval endpoints. Before starting, review trig foundations.
Method and conditions
Solve for the inner angle first. Write the repeated-angle families, map them back to the original variable, and retain only values inside the stated interval. Count shared endpoints once.
Worked example: show the decisions
For sin(2x)=1/2 on 0≤x<2π, the inner angle runs through two full cycles. Its solutions are π/6,5π/6,13π/6,17π/6. Dividing each by two gives x=π/12,5π/12,13π/12,17π/12.
Contrast: why a plausible shortcut fails
An inverse sine calculator returns one principal angle; it does not enumerate both sine branches or all periods. The interval length for 2x is twice the interval length for x.
Transfer: try before revealing the answer
How many solutions does cos(3x)=0 have on 0≤x<π? Explain without plotting every point.
Transfer solution and check
The inner angle ranges from 0 to 3π, excluding 3π. Zeros π/2,3π/2,5π/2 give three x-values. No endpoint is a cosine zero.
How many solutions does sin(x)=0 have on 0≤x≤3π?
- Zeros occur at integer multiples of π, including both endpoints zero and 3π.
- Using these quantities: 3+1 = 4.
- Check the original domain, requested quantity and units; do not substitute a related parameter.
Compare the other answer choices
Return only the principal zero.
Omit one included endpoint.
Assume two zeros per π-sized interval without removing duplicates.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus Course at a Glance