Math · Precalculus · Worked example

Convert between degrees and radians

Convert 150° to radians and 7π/4 to degrees. Then find the length of the arc that a 150° angle cuts from a circle of radius 12 cm.

Degrees to radians

Multiply by π/180 and simplify the fraction.

150 \cdot \frac{π}{180} = \frac{5 π}{6}

Multiply by 180/π: the π cancels.

\frac{7 π}{4} \cdot \frac{180^{\circ}}{π} = 315^{\circ}

s = rθ needs θ in radians, so use 5π/6, not 150.

\begin{array}{l} 12 \cdot \frac{5 π}{6} = 10 π \\ 10 π \approx 31.4 \end{array}

150° = 5π/6 and 7π/4 = 315°. The arc is 10π ≈ 31.4 cm long.

Convert 225° to radians and 5π/3 to degrees.

Show the answer and explanation

225 · π/180 = 5π/4, and 5π/3 · 180/π = 300.

225 \cdot \frac{π}{180} = \frac{5 π}{4}

Open the Unit Circle and set the angle to 150°: the heading reads θ = 5π/6, and the point is (−√3/2, 1/2), in quadrant II.

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