Convert Angles Using a Half-Turn Equivalence

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Convert 60° to radians exactly. Candidate answers: (A) 1/3π (B) 1/6π (C) 2/3π (D) 2π.
• Independent check inputs: Convert 90° to radians exactly. Candidate answers: (A) 1/2π (B) 1/4π (C) 1π (D) 3π.

Learn the idea

Radians measure an angle by arc length relative to radius. On the unit circle, cosine and sine are horizontal and vertical coordinates, so quadrant signs and special triangles support exact values beyond acute angles. A half turn is both 180 degrees and π radians. Multiplying by the matching conversion ratio changes the measurement unit while preserving the same geometric angle.

Here is the method on the supplied teaching case. Use 180°=π radians. Multiply 60 by π/180 to obtain (2/6)π. A unit conversion changes the number, not the angle.

Do not mix a degree value with a radian formula without conversion. Preserve exact multiples of π when requested; a calculator’s angle mode must match the unit of any entered angle.

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: Convert 270° to radians exactly. Candidate answers: (A) 3/2π (B) 3/4π (C) 3π (D) 9π. Show the mathematical reason, not only the selected answer. Expected result: 3/2π. Use 180°=π radians. Multiply 270 by π/180 to obtain (9/6)π. A unit conversion changes the number, not the angle.

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