BROCK UNIVERSITY MATHEMATICS MODULES
12B1.2: Relating Degrees and Radians
Author: Tyson Moss
WWW
• What it is: A formula for converting radians to degrees or the other way around.
• Why you need it: Given an angle in degrees, you may find it useful to convert it to radian
measure, perhaps for interpreting physical quantities. Also, some people understand
degrees more readily than radians, so converting from radians to degrees may help in
visualizing the size of an angle.
• When to use it: To convert degrees to radians, or radians to degrees.
PREREQUISITES
Before you tackle this module, make sure you have completed these modules:
Introduction to Radian Measure 12B1.1, Degree Measure, and Unit Conversion
WARMUP
Before you tackle this module, make sure you can solve the following exercises. If you have
difficulties, please review the appropriate prerequisite modules.
(Answers below.)
1. Determine the third angle of a triangle, given these two angles:
(a) A = 60◦ , B = 50◦
(b) A = 53◦ , B = 81◦
2. If you remove the following angle from a circle, how much is left?
(a) π radians (b)
π
3π
radians (c)
radians
2
4
3. Convert each speed into the specified units:
(a) 1 m/min into m/s (b) 7 cm/s into mm/s (c) 2 km/h into m/s
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4. Consider the arc s of a circle of radius r that subtends an angle θ. Calculate the unknown
quantity in each case.
(a) s = 8.1 m, r = ?, θ = 1.2 radians
(b) s = ?, r = 4.8 cm, θ = 1.6 radians
(c) s = 5 cm, r = 10.6 mm, θ = ?
3π
5π
1
Answers: 1.(a) 70◦ (b) 46◦ 2.(a) π radians (b)
radians (c)
radians 3.(a)
m/s (b) 70 mm/s
2
4
60
5
(c) m/s 4.(a) r = 6.75 m (b) s = 7.68 cm (c) θ ≈ 4.72 radians
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FOCUS QUESTION
To help you understand an important aspect of this lesson, focus your attention on this
question, which will be answered towards the end of the lesson.
You walk on a circular path, with radius 10 m, at a speed of 2 m/s. How long
does it take you to walk:
(a) a 30◦ portion of the path?
(b) once around the entire circular path?
Introduction
In Module 12B1.1, you learned what a radian is and how to use it. In this module you will
learn to convert radians to degrees and degrees to radians. This is potentially useful because most
people are more comfortable using degrees, but in some applications it is important to use radians
(in order to easily interpret various units, or because derivatives of trigonometric functions are
involved).
As we discussed in Module 12B1.1, the angle for a complete circle can be expressed as 360◦ , or
equivalently as 2π radians. In other words,
360◦ = 2π radians
Of course, just because an angle of 360◦ is the same as an angle of 2π radians, the number 2π is
NOT the same number as the number 360. But this is just the same as 10 mm = 1 cm; 10 and 1
are not equal as numbers, but 10 mm is the same length as 1 cm. It’s important to pay attention
to units.
Because an angle of 0◦ is the same as an angle of 0 radians, the degree measure and the radian
measure of an angle are proportional. This is the same situation for many other unit conversions,
such as mm to cm, km to miles, and so on. An exception is the process for converting between
Fahrenheit and Celsius degrees; because 0◦ F is not the same as 0◦ C, the number of Fahrenheit
degrees and the number of Celsius degrees are not proportional, and converting from one to the
other is a little more complicated.
Using the proportionality between them, we can figure out a nice little formula for converting
radians to degrees and visa versa. Let’s think about this a bit first.
Say you have an angle of 1◦ . To make a full circle, you would need 360 single degrees. In other
words, a full circle is 360 · 1◦ . This is the same as 360◦ . And we could say the same about radians:
2π · 1 rad = 2π rad.
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If we use these new observations in the equation 360◦ = 2π and, say, solve for 1◦ , we get:
360 · 1◦ = 2π rad
2π
1◦ =
rad
360
π
1◦ =
rad
180
We can see that 1◦ is equivalent to
π
radians. This gives us a nice conversion factor.
180
DEFINITION
To convert from degrees to radians, use the following conversion factor.
1◦ =
π
rad
180
To illustrate this, suppose you want to convert an angle measure of 23◦ to radians. Well, all
you do is multiply both sides of the conversion factor by 23, which gives you:
π
rad
180
≈ 0.4 rad
23◦ = 23 ×
Pretty good stuff, eh? Now that we know how to convert degrees to radians, how about the
other way?
π
Well, just rearrange the equation 1◦ =
rad to solve for 1 radian. See if you can do it on
180
your own; the final equation should be:
1 rad =
180 ◦
π
DEFINITION
To convert from radians to degrees, use the following conversion factor.
1 rad =
180◦
π
So thinking in the same way, if you want to convert an angle measure of 2 radians to degrees,
180 ◦
multiply both sides of the equation 1 rad =
by 2. This gives you:
π
180 ◦
π
◦
≈ 114.6
2 rad = 2 ×
Do you understand how to do these conversions? Try some on your own to make sure!
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PRACTICE
(Answers below.)
1. Convert the following degree measures into radians:
(a) 30◦ (b) 90◦ (c) 101◦ (d) 24◦
2. Convert the following radian measures into degrees:
(a) π radians (b)
Answers:
(d) 60.16
1.(a)
◦
3π
π
radians (c)
radians (d) 1.05 radians
4
8
π
π
2π
radians (b)
radians (c) 1.763 radians (d)
radians 2.(a) 180◦
6
2
15
(b) 45◦
(c) 67.5◦
Now let’s look at an applied example where such conversions are useful.
EXAMPLE 1
Stephanie cycles around a circular track at a speed of 15 m/s. She stays the same distance from
the centre of the track while she cycles through an angle of 45◦ . If it takes her 5 s to cycle this
distance, how far from the centre of the track is she?
SOLUTION
s
only works when the angle is measured in
r
π
radians. This means we’ll need to use the formula 1◦ =
rad to convert Stephanie’s angle 45◦
180
to radians. As we did earlier, we can do this by multiplying both sides of the conversion formula
by 45, as follows:
Recall from Module 12B1.1 that the formula θ =
π
rad
180
45π
45◦ =
rad
180
π
= rad
4
≈ 0.79 rad
1◦ =
π
radians. Now that we know the value of θ in radians, we
4
s
would like to use the equation θ = to determine the radius of the circular arc that Stephanie
r
cycles on. However, we don’t know s; calculating this value is our next task. This can be
computed by multiplying how fast Stephanie cycles by the time she takes. Doing this, we get:
This means that 45◦ is the same as
4
s = speed · time
= 15 m/s · 5 s
= 75 m
s
Finally, we can use the formula θ = to determine the radius of the circular arc that Stephanie
r
is travelling on:
s
r
s
r=
θ
75
=
π/4
300
=
π
≈ 95.5 m
θ=
300
This means the radius of the path Stephanie cycles on is
m, which is approximately 95.5
π
m.
Now it’s time to try a few applied problems on your own.
PRACTICE
(Answers below.)
3. Ski lifts carry people up the side of a hill at a constant rate. The rate depends on the
slope of the hill. A ski-lift-ologist decided to survey how long it takes to carry someone
from the bottom of the hill to the top for various ski lifts. The table below gives the
distance the people travelled, how long it took to get there, and the angle of the hill, for
two different ski lifts.
Length Of Trip (m)
45 m
75 m
Time (s)
30 s
75 s
Angle of the Hill
45◦
60◦
After a magazine editor noticed his survey, the ski-lift-ologist was asked to write an
article including this information, subject to a few conditions. He was asked to convert
all the angles to radians,a and also to include data for two more ski lifts at the hill.
Assuming the relationship between angle and speed is linear, complete a chart like the
one above, making sure that radian measure is used. Include extra rows for the angles
π
4π
of
radians and
radians, where the lengths of the trips will be 100 m. Finally add
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9
an extra column and calculate the speed of each trip in metres per second.
a
Was the editor trying to make the article as high-brow as possible?
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Answers:
Length of trip (m)
Time (s)
100 m
50 s
45 m
30 s
75 m
75 s
100 m
300 s
Angle of the Hill
Speed (m/s)
π
6
π
4
π
3
4π
9
2 m/s
1.5 m/s
1 m/s
1
m/s
3
Now, let’s solve the focus question from above, if you haven’t already done so.
RECAP OF FOCUS QUESTION
Recall the focus question, which was asked earlier in the lesson.
You walk on a circular path, with radius 10 m, at a speed of 2 m/s. How long
does it take you to walk:
(a) a 30◦ portion of the path?
(b) the entire circular path?
SOLUTION
First let’s write down what we know:
(i) r = 10 m
(ii) v = 2 m/s
(iii) In Part (a), θ = 30◦ , and in Part (b), θ = 360◦ .
There are two parts to this question: We need to calculate the time it takes for you to walk
the 30◦ portion of the path, and to walk the entire path, making a full circle.
One way to solve this problem is to calculate how long it takes you to walk the entire circle
first. Then, knowing that 30◦ is 1/12 of a full circle, we can determine the time needed to
walk the 30◦ portion of the path just by dividing the time needed to walk the full circle by 12.
There are other ways to attack this problem, so you might like to explore them and see if you
can obtain the same result.
Calculating the time to walk the entire circle isn’t too bad. What we first need to do is recall
that a full circle has 2π radians. This will allow us to calculate the arc length (circumference)
of the circle:
s
r
s = rθ
θ=
= 10 m · 2π
= 20π m
Therefore, the arc length of the circle is 20π m.
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Now, we need to calculate how long it takes for you to walk the full circle. We know your
speed is 2 m/s, and to obtain the total time we can divide the length of your journey by your
speed, as follows:
distance = speed · time
distance
time =
speed
s
t=
v
20π m
=
2 m/s
= 10π s
≈ 31.4 s
It takes you about 31.4 s to walk around the entire circle.
Knowing that 30◦ is 1/12 of a full circle, we just have to divide the time it takes you to walk
around the full circle by 12 to determine how long it takes you to walk the 30◦ portion of the
path:
t
10π s
=
12
12
5π
=
s
6
≈ 2.6 s
This means it takes you about 2.6 s to walk the 30◦ portion of the path.
KEY IDEA
Let’s go back and look at the question involving Stephanie cycling around a track. We found
that the arc that she cycles on is 75 m long and the corresponding angle is 45◦ . If we don’t
convert the angle to radians, let’s see what happens:
s
r
s
r=
θ
75 m
=
45◦
5
= m/◦
3
θ=
Hold on there, cowboy some of you may be yelling right now. What in the WORLD is a m/◦?
We were expecting the formula to produce a value for how far Stephanie is from the centre of
the circle, but the unit is just so bizarre that it’s hard to make sense of it. This leaves us with
no idea how long the radius of this circle is. This is why we convert the angle measure into
radians; being unitless, this makes the final distance come out in a unit (metres) that we can
understand.
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INVESTIGATION
Explore this on your own!
For this exploration, a calculator will be helpful.
What you should do is change your calculator from degree mode to radian mode (denoted on
most calculators by deg and rad). Once you do this, input a few angles into the sin(x) function
(for example, sin(π)). See if you can draw one period of the sine function, and find where the
maximum and minimum values are, as well as where the graph crosses the x-axis in terms of
radians.
After that, perform the same task for the cos(x) function.
WWW
• What we did: We converted angle measures from degrees to radians, and from radians
to degrees.
• Why we did it: This helps you get a sense for how large a radian is, as many people
have this sense for degrees but not for radians. It’s also helpful for interpreting units in
applied problems.
• What’s next: The next few modules use what you’ve learned about angles to define and
study trigonometric functions, such as sine, cosine, and tangent.
EXERCISES
1. Convert the following angles from degrees to radians.
(a) 90◦
(b) 60◦
(c) 253◦
(d) 18◦
(e) 191◦
2. Convert the following angles from radians to degrees.
(a) π radians
π
(b)
radians
2
(c) 1.25 radians
(d) 2π radians
5π
(e)
radians
3
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3. Eric loves to go to the carnival, but he only attends to go to one attraction; the funhouse!
He enjoys climbing the revolving “barrels” throughout them. He sits as still as possible
and lets the machine rotate the barrel as he tries to hold on as long as possible until he
slides back down.
If he is able to hold on for an angle of 60◦ and the radius of the barrel is 1.5 m, how far
does Eric move before he starts to slide?
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