If θ is an acute angle and sin(θ + 18°) = 1/2, then what is the value of θ in radians (circular measure)? Use the principal acute-angle solution for sine and convert the final angle from degrees to radians.

Introduction / Context: This question tests solving a simple trigonometric equation involving sine and then converting degrees to radians. The key idea is recognizing when sin of an angle equals 1/2. The standard angles where sine equals 1/2 are 30° and 150° (within 0° to 180°). Because θ is given as an acute angle, θ lies between 0° and 90°. That means θ + 18° lies between 18° and 108°. In that range, the sine value 1/2 corresponds to 30° (not 150°). After finding θ in degrees, convert it to radians using the conversion factor radians = degrees * (π/180). Many mistakes happen when students choose the wrong sine angle (150°) or forget to subtract 18° after matching the sine value.

Given Data / Assumptions:

• θ is acute: 0° < θ < 90° • sin(θ + 18°) = 1/2 • Standard values: sin 30° = 1/2 • Required: θ in radians

Concept / Approach: First find the angle α = θ + 18°. Since sin α = 1/2 and α must lie between 18° and 108°, select α = 30°. Then θ = 30° − 18° = 12°. Convert 12° to radians using π/180 per degree.

Step-by-Step Solution: 1) Let α = θ + 18°. 2) Given sin α = 1/2. 3) In the range relevant to an acute θ, take α = 30° (since sin 30° = 1/2). 4) Solve for θ: θ + 18° = 30° ⇒ θ = 12° 5) Convert 12° to radians: θ = 12 * (π/180) = π/15

Verification / Alternative check: Check by substitution: θ = 12° gives θ + 18° = 30°, and sin 30° = 1/2, which matches the condition exactly. Also, π/15 radians equals 12° because (π/15) * (180/π) = 12, confirming the conversion is consistent.

Why Other Options Are Wrong: • π/12 corresponds to 15°, which would make θ + 18° = 33°, not a sine value of 1/2. • 2π/5 corresponds to 72°, too large for the computed θ. • 3π/13 is not a standard conversion here and does not satisfy the equation. • π/18 corresponds to 10°, which would give θ + 18° = 28°, not 30°.

Common Pitfalls: • Choosing 150° instead of 30° for sin = 1/2, ignoring the acute constraint. • Forgetting to subtract 18° after identifying the sine angle. • Converting degrees to radians using the wrong factor (mixing 180/π and π/180).

Final Answer: π/15