Radians & Degrees
Aim: By the end of this note, you will be able to transform degrees to radians and vice versa.
 You probably tell me the answer is 360^{o} (degrees). Is this the only unit for measuring angle? Before we answer that question, let us first look into the history of why a circle has 360^{o} (degrees).
 The answer lies with the ancient Babylonians. They used a base 60 number
system. Part of its legacy is still with us. We say 1 minute = 60 seconds
and 60 minutes = 1 hours.
In notations these are 3600 '' (seconds) = 60 ' (minutes) = 1 ^{o} (degree in this context).  The ancient Babylonians also had 360 days in a year. So every 360 days, they "experienced" a cylce or rotation (winterspringsummerautumnwinter). Thus, a circle is logically defined as a rotation that has 360^{o} (degree). A degree is then defined as 1/360 of a full rotation. But this definition is not very satisfactory to mathematicians. Instead of 360 parts, why not divide a rotation into 100 parts or even 400 parts? They would like to have a definition that depends only on the characteristics of a circle.
 Here is that attempt.
A radian is defined as the size of the angle such that the length of the arc, part of the circumference that belongs to sector \( \theta \), is the same as the radius. Sector is the shaded area in this diagram.
\[ \begin{align} \large 1^c = \frac{180^o}{\pi} \\ \large \pi^c = 180^o \end{align} \]
Why is 1 ^{c} = 180^{o}/π?
We know that the circumference of a complete circle (\(C\)) is 2πr.
The arc length (\(s\)) of a sector with angle θ is thus
\( s = (\frac{\theta}{360^o})2\pi r \)
If θ = 1 radian then \( s = r \).
Thus we have,
\( r = (\frac{1 radian}{360^o})2\pi r \)
\( 1 = (\frac{1 radian}{180^o})\pi \)
\( 1 radian = \frac{180^o}{\pi} \)
\( 1 radian \approx 57.3^o \) ( 3 s.f.)
Example 1.Convert the following angles
to radians. (i) 85^{o}12'18" (ii) 260^{o}12" Solutions: (i) 85^{o}12'18" = [ 85+ (12/60) + (18/3600) ]^{o} ; we first change the whole angle into a decimal expression in degree 85.205^{o} = 85.205(π/180^{o}) radian ≈ 1.49 radian (3 s.f.) (ii) 260^{o}15" = [ 260 + (15/3600) ]^{o}(π/180^{o}) radian ≈ 4.54 radian (3 s.f) 
Example 2.Express the following angles in terms
of degrees and minutes: (i) 2.53^{c} (ii) 5(π/3) radian Solutions: (i) 2.53^{c} = 2.53^{c}(180^{o}/π^{c}) = 144.9583222^{o} (from calculator) ≈144^{o}57' [to the nearest minute] (ii) 5(π/3) radian = 5(π/3)^{c} (180^{o}/π^{c}) = 5(180^{o})/3 = 300^{o} 
Summary:
From θ radian to degree: \( \theta^c \left( \frac{180^o}{\pi^c} \right) \)From θ degree to radian: \( \theta^o \left( \frac{\pi^c}{180^o} \right) \)
A way to remember the radiandegree conversion formula. Let us study the two formula above. θ represents the angle that you need to convert, so it must be in the formula. After that we have 180^{o} and π^{c} to fit into our formula.
If we need the answer to be in radian then we will have π^{c} in the numerator and 180^{o} in the denominator. 
Worksheet
Unless otherwise specified in the question, report your answers exactly or to 3 significant figures.
 Complete the table below.
0^{o}60^{o}180^{o}270^{o}30^{o}576^{o}360^{o}0^{c}π^{c}/8π^{c}2π^{c}/3π^{c}/162π^{c}
 Convert the following angles to radian:
 45^{o}
 17^{o} 18'
 164^{o }2' 30"
 Convert the following angles to degree:
 π^{c}/7
 0.59^{c}
 4.3^{c }

Assume that we have a circular pizza as in figure 1.
Let the radius of this pizza be 10 cm and θ = 0.8 radian. Convert the angle θ = 0.8 radian to degree.
Report your answer to 2 decimal places.
 How large is the angle θ as a percentage of
the total circular pizza?
 You are given the sector AOB to enjoy and you especially love the crispy
rim of pizza.
Use your result in (b) above or otherwise to find the amount of crispy rim ACB that you can enjoy? Report your answer to 1 significant figure.
[Hint: Arc ACB is a part of the circumference 2πr.](Generalization)
 Instead of θ =0.8 radian, we will now have
a general angle θ = x^{o}.
Find an expression for arc ACB in term of x^{o}, π and r.  Instead of measuring θ in degree we now measure
the general angle θ in radian. Let this angle
be θ^{c}. Find an expression for arc
ACB in term of θ^{c}, π
and r.
 Compare your answers for (d) and (e), and decide which expression for
arc ACB is more simple?
 Convert the angle θ = 0.8 radian to degree.
Report your answer to 2 decimal places.
 Let
us assume that Hong Kong (H) and Kaifeng (K), China are located on the same
longitude as in figure 2. Hong Kong is located on northern latitude 22^{o}
15' and Kaifeng is on northern latitude 34^{o} 36'. We further assume
that the earth is a sphere with a radius of 6378 km. Let O be the center of
a spherical earth and OE a part of the equator (i.e. latitude 0^{o}).
 Find the angle θ in degree accurate to 2 decimal
places. [Hint: the angle between HO and OE is 22^{o} 15'.]
 Convert your answer in (a) to radian accurate to 3 decimal places.
 Use your result in (b) or otherwise to calculate the distance from Hong
Kong to Kaifeng to the nearest km. [Hint: refer to your answers for question
4 above.]
 Find the angle θ in degree accurate to 2 decimal
places. [Hint: the angle between HO and OE is 22^{o} 15'.]
Answers:
1) 22.5^{o}, π^{c}/3, 120^{o},
3p^{c}/2, p^{c}/6,
11.25^{o}, 3.2p^{c} (2a)π^{c}/4
= 0.785 rad (2b) 0.302 radian (3c) 2.86 radian (3a) 25.7^{o} (3b) 33.8^{o
} (3c) 246^{o} (4a) 45.84^{o} (4b) 12.7% (4c) 8 (4d) (x/360)2πr
(4e) θ^{c}r (4f) the one with radian is more
simple. (5a) 12.35^{o} (5b) 0.216 radian (5c) 1378 km [1375 km if you
use 12.35^{o}(π/180^{o})(6378)]