Radians and degrees Problem set B Worked solutions

Fill in the missing entries of this table, writing radians as fractions of π.

• \(\frac{4\pi}{3}\times\frac{180}{\pi}=240^\circ\)
• \(\frac{7\pi}{6}\times\frac{180}{\pi}=210^\circ\)
• \(\frac{\pi}{3}\times\frac{180}{\pi}=60^\circ\)
• \(300^\circ\times\frac{\pi}{180}=\frac{5\pi}{3}\)
• \(135^\circ\times\frac{\pi}{180}=\frac{3\pi}{4}\)
• \(315^\circ\times\frac{\pi}{180}=\frac{7\pi}{4}\)

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

1. (a) −210°

   • \(-210^\circ\times\frac{\pi}{180}=-\frac{7\pi}{6}\)

2. (b) 480°

   • \(480^\circ\times\frac{\pi}{180}=\frac{8\pi}{3}\)

3. (c) −495°

   • \(-495^\circ\times\frac{\pi}{180}=-\frac{11\pi}{4}\)

4. (d) 800°

   • \(800^\circ\times\frac{\pi}{180}=\frac{40\pi}{9}\)

Write each angle in degrees.

1. (a) \(\frac{3\pi}{10}\)

   • \(\frac{3\pi}{10}\times\frac{180}{\pi}=54^\circ\)

2. (b) \(-\frac{11\pi}{4}\)

   • \(-\frac{11\pi}{4}\times\frac{180}{\pi}=-495^\circ\)

3. (c) \(\frac{17\pi}{6}\)

   • \(\frac{17\pi}{6}\times\frac{180}{\pi}=510^\circ\)

4. (d) \(-\frac{29\pi}{12}\)

   • \(-\frac{29\pi}{12}\times\frac{180}{\pi}=-435^\circ\)

Give each answer to 3 significant figures.

1. (a) 55° in radians

   • \(55^\circ\times\frac{\pi}{180}=0.960\)

2. (b) 3.4 radians in degrees

   • \(3.4\times\frac{180}{\pi}=195^\circ\)

3. (c) −2.5 radians in degrees

   • \(−2.5\times\frac{180}{\pi}=−143^\circ\)

A record turntable spins through 990°.

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

   • \(990^\circ\times\frac{\pi}{180}=\frac{11\pi}{2}\)

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

   • \(\frac{990}{360}=2.75\)

A hamster wheel has radius 12 cm. A sticker on the edge moves 51 cm along the edge. Find

1. (a) the angle turned, in radians

   • \(\theta=\frac{51}{12}=4.25\)

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

   • \(4.25\times\frac{180}{\pi}=244^\circ\)

The hour hand of a clock moves for 7 hours. Find

1. (a) the angle in degrees

   • One full turn is 12 hours, so the hand turns 7/12 of 360° = 210°.

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

   • \(210^\circ\times\frac{\pi}{180}=\frac{7\pi}{6}\)

Write 3.8 radians, \(\frac{7\pi}{6}\) radians and 215° in order, smallest first.

• Change all three to degrees.
• \(\frac{7\pi}{6}\) radians = 210°; 215°; 3.8 radians = 217.7°
• Put them in order: \(\frac{7\pi}{6}\), 215°, 3.8 radians

A roundabout turns through −13 radians. Find

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

   • \(−13\times\frac{180}{\pi}=−744.8^\circ\)

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

   • 744.8 = 2 × 360 + 24.8. The minus sign means clockwise: 2 full turns and a bit more.