Angles and Radian Measure: Quick Revision

A brief note on Angles and Radian Measure. Read it before a test. Try each question at the end before you open its answer.

Angles as turns

• An angle is a turn from the initial side to the terminal side. Anticlockwise is positive, clockwise is negative.
• One revolution is $360^\circ$. Angles that differ by a multiple of $360^\circ$ are coterminal: they end in the same place.
• An angle lies in the quadrant of its terminal side. An angle on an axis is quadrantal.
• $1^\circ = 60'$ and $1' = 60''$, so a minute is a sixtieth of a degree, not a hundredth.

Radians

• One radian is the angle at the centre when the arc is as long as the radius:
  • $\theta = \dfrac{s}{r}$
• A full turn is $2\pi$ rad, so $\pi$ rad $= 180^\circ$, and $1$ rad is about $57.3^\circ$.
• Degrees into radians: multiply by $\dfrac{\pi}{180}$. Radians into degrees: multiply by $\dfrac{180}{\pi}$.
• An angle with no unit is in radians. Python's math functions use radians.

Arcs, sectors and speed

• The arc of a circle, with $\theta$ in radians:
  • $s = r\theta$
• The area of a sector:
  • $A = \dfrac{1}{2}r^2\theta = \dfrac{1}{2}rs$
• A rolling wheel moves $2\pi r$ in each turn.
• For $n$ revolutions each minute, the angular speed and the rim speed are:
  • $\omega = \dfrac{2\pi n}{60}$ rad/s
  • $v = r\omega$

A worked example

A scooter wheel of radius $0.3$ m makes $7$ turns each second. Find its angular speed, and the speed of the scooter in km/h.

1. The angle each second. Each turn is $2\pi$ rad, so $\omega = 7 \times 2\pi = 14\pi \approx 43.98$ rad/s.
2. The speed. $v = r\omega = 0.3 \times 43.98 \approx 13.19$ m/s.
3. In km/h. Multiply by $3.6$: about $47.5$ km/h.

Check: in one second the wheel rolls $7 \times 2\pi \times 0.3 \approx 13.19$ m. Both ways agree.

Practice with new numbers

A program to try

The program finds the smaller angle between the hands of a clock at four times, in degrees and as a multiple of $\pi$. The hour hand moves half a degree each minute, and the minute hand moves six degrees.

Mistakes to avoid

• Losing the sign. A clockwise turn is negative. A quarter turn clockwise is $-90^\circ$, which ends like $270^\circ$.
• The wrong fraction. Degrees into radians is $\times\frac{\pi}{180}$, not $\times\frac{180}{\pi}$.
• Degrees in $s = r\theta$. The formulas for the arc, the sector and $v = r\omega$ need radians.
• Revolutions as radians. One revolution is $2\pi$ rad. Divide by $60$ to go from each minute to each second.
• Losing the half. The area of a sector is $\frac{1}{2}r^2\theta$.

Check yourself

1. Which angle from $0^\circ$ to $360^\circ$ ends where $-610^\circ$ ends, and in which quadrant does it lie?

2. Write $\dfrac{5\pi}{9}$ rad in degrees, and $48^\circ 45'$ in decimal degrees.

3. A swing seat hangs on ropes $2.4$ m long and moves along an arc of $1.8$ m. Through what angle does it swing?

4. A flower bed is a sector of radius $15$ m with an angle of $\dfrac{\pi}{4}$ at the centre. How much ground does it cover?

5. A record player turns at $45$ revolutions per minute. A point on the edge of the record is $15$ cm from the centre. How fast does it move?