Answers: Introduction to radians
These are the answers to Questions: Introduction to radians.
Please attempt the questions before reading these answers!
Q1
1.1. Multiplying \(30^\circ\) by \(\pi\) and dividing by \(180\) gives \(\dfrac{30\pi}{180} \textsf{ rad} = \dfrac{\pi}{6} \textsf{ rad} = 0.524 \textsf{ rad}\) to three decimal places.
1.2. Multiplying \(105^\circ\) by \(\pi\) and dividing by \(180\) gives \(\dfrac{105\pi}{180} \textsf{ rad} = \dfrac{7\pi}{12} \textsf{ rad} = 1.833 \textsf{ rad}\) to three decimal places.
1.3. Multiplying \(298^\circ\) by \(\pi\) and dividing by \(180\) gives \(\dfrac{298\pi}{180} \textsf{ rad} = \dfrac{149\pi}{90} \textsf{ rad} = 5.201 \textsf{ rad}\) to three decimal places.
1.4. Multiplying \(61^\circ\) by \(\pi\) and dividing by \(180\) gives \(\dfrac{61\pi}{180} \textsf{ rad} = 1.064 \textsf{ rad}\) to three decimal places.
1.5. Multiplying \(353^\circ\) by \(\pi\) and dividing by \(180\) gives \(\dfrac{353\pi}{180} \textsf{ rad} = 6.161 \textsf{ rad}\) to three decimal places.
1.6. Multiplying \(197^\circ\) by \(\pi\) and dividing by \(180\) gives \(\dfrac{197\pi}{180} \textsf{ rad} = 3.438 \textsf{ rad}\) to three decimal places.
Q2
2.1. Multiplying \(\dfrac{\pi}{3} \textsf{ rad}\) by \(180\) and dividing by \(\pi\) gives \(\dfrac{180\pi}{3\pi} ^\circ = 60 ^\circ\).
2.2. Multiplying \(\dfrac{2\pi}{3}\textsf{ rad}\) by \(180\) and dividing by \(\pi\) gives \(\dfrac{360\pi}{3\pi} ^\circ = 120 ^\circ\).
2.3. Multiplying \(\dfrac{\pi}{7} \textsf{ rad}\) by \(180\) and dividing by \(\pi\) gives \(\dfrac{180\pi}{7\pi} ^\circ = 25.714 ^\circ\) to three decimal places.
2.4. Multiplying \(\dfrac{5\pi}{7}\textsf{ rad}\) by \(180\) and dividing by \(\pi\) gives \(\dfrac{900\pi}{7\pi} ^\circ = 128.571 ^\circ\) to three decimal places.
2.5. Multiplying \(5\textsf{ rad}\) by \(180\) and dividing by \(\pi\) gives \(\dfrac{900}{\pi} ^\circ = 286.479 ^\circ\) to three decimal places.
2.6. Multiplying \(\dfrac{3}{4} \textsf{ rad}\) by \(180\) and dividing by \(\pi\) gives \(\dfrac{540}{4\pi} ^\circ = \dfrac{135}{\pi} ^\circ = 42.972 ^\circ\) to three decimal places.
Q3
3.1. In this case, the length of the arc is \(\dfrac{7\pi}{8} = 2.749\) (to 3dp) and the area of the sector is \(\dfrac{49\pi}{16} = 9.621\) (to 3dp).
3.2. In this case, the length of the arc is \(\dfrac{\pi}{2} = 1.571\) (to 3dp) and the area of the sector is \(\dfrac{\pi}{12} = 0.262\) (to 3dp).
3.3. In this case, the length of the arc is \(14\pi = 43.982\) (to 3dp) and the area of the sector is \(\dfrac{525\pi}{2} = 824.668\) (to 3dp).
Version history and licensing
v1.0: initial version created 08/23 by Ifan Howells-Baines, Mark Toner as part of a University of St Andrews STEP project.
- v1.1: edited 05/24 by tdhc.