Hint
The reflex angle is 360° minus the minor angle. An additional revolution adds 2π radians to the swept angle.
Worked solution
(a)
The minor angle is \(\displaystyle \frac{144\pi }{180}= \frac{4\pi }{5}\). The reflex angle is \(\displaystyle 2\pi - \frac{4\pi }{5}= \frac{6\pi }{5}\) rad.
Answer: Minor \(\displaystyle \frac{4\pi }{5}\); reflex \(\displaystyle \frac{6\pi }{5}\).
(b)
Use the minor angle: \(\displaystyle s= r\theta = 5(\frac{4\pi }{5})= 4\pi \mathrm{cm}\).
Answer: \(4\pi\text{ cm}\).
(c)
The swept angle is \(\displaystyle \frac{6\pi }{5}+ 2\pi = \frac{16\pi }{5}\), which exceeds one full turn. The travelled distance is \(\displaystyle 5(\frac{16\pi }{5})= 16\pi \mathrm{cm}\). Reducing the angle to its final direction would lose one revolution of travel.
Answer: \(\displaystyle \frac{16\pi }{5}\) rad; \(16\pi\text{ cm}\).
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 2 marks
- 1 mark: Convert the minor angle to \(\displaystyle \frac{4\pi }{5}\) rad.
- 1 mark: Give the reflex angle \(\displaystyle \frac{6\pi }{5}\) rad.
Alternative methods: Converting the reflex angle 216° directly is accepted.
Common error: Using 144 as a radian angle.
(b) · 2 marks
- 1 mark: Use arc length \(\displaystyle r\theta\) with the minor angle in radians.
- 1 mark: Obtain exactly 4π cm.
Alternative methods: Taking \(\displaystyle \frac{144}{360}\) of the circumference is accepted.
Common error: Using the reflex angle for the minor arc.
(c) · 3 marks
- 1 mark: Include the additional full turn in the total angle.
- 1 mark: Obtain total angle \(\displaystyle \frac{16\pi }{5}\) rad.
- 1 mark: Obtain distance 16π cm.
Alternative methods: Adding the reflex arc length 6π cm to the circumference 10π cm is accepted, with the swept angle stated.
Common error: Reducing the swept angle modulo 2π before finding travelled distance.
Total: 7 marks.