Radians and degrees Problem set A One question at a time

Copy and complete the table. Give radians as multiples of π.

• \(\frac{5\pi}{4}\times\frac{180}{\pi}=225^\circ\)
• \(\frac{11\pi}{6}\times\frac{180}{\pi}=330^\circ\)
• \(\frac{2\pi}{3}\times\frac{180}{\pi}=120^\circ\)
• \(150^\circ\times\frac{\pi}{180}=\frac{5\pi}{6}\)
• \(210^\circ\times\frac{\pi}{180}=\frac{7\pi}{6}\)
• \(315^\circ\times\frac{\pi}{180}=\frac{7\pi}{4}\)

Write each angle in radians, as a multiple of π.

1. (a) −150°

   • \(-150^\circ\times\frac{\pi}{180}=-\frac{5\pi}{6}\)

2. (b) 420°

   • \(420^\circ\times\frac{\pi}{180}=\frac{7\pi}{3}\)

3. (c) −585°

   • \(-585^\circ\times\frac{\pi}{180}=-\frac{13\pi}{4}\)

4. (d) 1000°

   • \(1000^\circ\times\frac{\pi}{180}=\frac{50\pi}{9}\)

Write each angle in degrees.

1. (a) \(\frac{2\pi}{5}\)

   • \(\frac{2\pi}{5}\times\frac{180}{\pi}=72^\circ\)

2. (b) \(-\frac{7\pi}{4}\)

   • \(-\frac{7\pi}{4}\times\frac{180}{\pi}=-315^\circ\)

3. (c) \(\frac{13\pi}{3}\)

   • \(\frac{13\pi}{3}\times\frac{180}{\pi}=780^\circ\)

4. (d) \(-\frac{31\pi}{12}\)

   • \(-\frac{31\pi}{12}\times\frac{180}{\pi}=-465^\circ\)

Give each answer to 3 significant figures.

1. (a) 75° in radians

   • \(75^\circ\times\frac{\pi}{180}=1.31\)

2. (b) 2.1 radians in degrees

   • \(2.1\times\frac{180}{\pi}=120^\circ\)

3. (c) −4 radians in degrees

   • \(−4\times\frac{180}{\pi}=−229^\circ\)

A fairground wheel turns through 1170°.

1. (a) Write the angle in radians, as a multiple of π.

   • \(1170^\circ\times\frac{\pi}{180}=\frac{13\pi}{2}\)

2. (b) How many complete turns is this, as a decimal?

   • \(\frac{1170}{360}=3.25\)

A cyclist's wheel has radius 35 cm. A mark on the rim moves 98 cm along the rim. Find

1. (a) the angle turned, in radians

   • \(\theta=\frac{98}{35}=2.80\)

2. (b) the angle turned, in degrees (3 s.f.)

   • \(2.80\times\frac{180}{\pi}=160^\circ\)

The minute hand of a clock moves for 25 minutes. Find

1. (a) the angle in degrees

   • One full turn is 60 minutes, so the hand turns 25/60 of 360° = 150°.

2. (b) the angle in radians, in terms of π

   • \(150^\circ\times\frac{\pi}{180}=\frac{5\pi}{6}\)

Write 220°, 3.9 radians and \(\frac{6\pi}{5}\) radians in order, smallest first.

• Change all three to degrees.
• \(\frac{6\pi}{5}\) radians = 216°; 220°; 3.9 radians = 223.5°
• Put them in order: \(\frac{6\pi}{5}\), 220°, 3.9 radians

A turntable turns through −7.5 radians. Find

1. (a) the angle in degrees (3 s.f.)

   • \(−7.5\times\frac{180}{\pi}=−429.7^\circ\)

2. (b) the number of full turns, and the direction

   • 429.7 = 1 × 360 + 69.7. The minus sign means clockwise: 1 full turn and a bit more.