Find Every Solution in a Stated Cycle

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Find all solutions to sin θ=1/2 on 0π≤θ<2π. Candidate answers: (A) 1/3π and 2/3π (B) 1/6π and 5/6π (C) 1/6π and 7/6π (D) 1/6π only.
• Independent check inputs: Find all solutions to sin θ=1/2 on 2π≤θ<4π. Candidate answers: (A) 13/3π and 14/3π (B) 13/6π and 17/6π (C) 13/6π and 19/6π (D) 13/6π only.

Learn the idea

An identity holds for all inputs in its domain, while an equation asks for particular angle solutions. Algebraic identities, angle formulas and geometric laws must retain signs, intervals and their defining conditions. A positive sine level can occur twice in a full cycle, once on each side of its maximum. Reference-angle solutions repeat after adding whole multiples of two π.

Here is the method on the supplied teaching case. Within one cycle, sine 1/2 occurs at π/6 and 5π/6. Add the cycle shift 0π to both. The half-open interval contains both and avoids repeating an endpoint.

The specified interval limits which repeated solutions belong in the answer. Keep both within-cycle solutions and respect half-open endpoints; an inverse-sine calculator value alone usually gives only the principal branch.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Find all solutions to sin θ=1/2 on 14π≤θ<16π. Candidate answers: (A) 85/3π and 86/3π (B) 85/6π and 89/6π (C) 85/6π and 91/6π (D) 85/6π only. Show the mathematical reason, not only the selected answer. Expected result: 85/6π and 89/6π. Within one cycle, sine 1/2 occurs at π/6 and 5π/6. Add the cycle shift 14π to both. The half-open interval contains both and avoids repeating an endpoint.

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