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Section 6.1 Degree and Radian Measure (TR1)
Objectives
Convert between degrees and radians. Draw angles in standard position.
Subsection 6.1.1 Activities
Definition 6.1.1 .
An
angle is formed by joining two rays at their starting points. The point where they are joined is called the
vertex of the angle. The measure of an angle describes the amount of rotation between the two rays.
Activity 6.1.2 .
An angle that is rotated all the way around back to its starting point measures
\(360^\circ\text{,}\) like a circle. Use this to estimate the measure of the given angles.
(a)
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
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X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
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K
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Y
Stop sound output
O
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P
Repeat sonification output
\(\displaystyle 45^{\circ}\)
\(\displaystyle 90^{\circ}\)
\(\displaystyle 135^{\circ}\)
\(\displaystyle 180^{\circ}\)
(b)
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
\(\displaystyle 45^{\circ}\)
\(\displaystyle 90^{\circ}\)
\(\displaystyle 135^{\circ}\)
\(\displaystyle 180^{\circ}\)
(c)
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
\(\displaystyle 45^{\circ}\)
\(\displaystyle 90^{\circ}\)
\(\displaystyle 135^{\circ}\)
\(\displaystyle 180^{\circ}\)
Definition 6.1.3 .
An angle is in
standard position if its vertex is located at the origin and its initial side extends along the positive
\(x\) -axis.
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
An angle measured counterclockwise from the initial side has a positive measure, while an angle measured clockwise from the initial side has a negative measure.
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Activity 6.1.4 .
Estimate the measure of the angles drawn in standard position.
(a)
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
\(\displaystyle 45^{\circ}\)
\(\displaystyle 90^{\circ}\)
\(\displaystyle 135^{\circ}\)
\(\displaystyle 180^{\circ}\)
(b)
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
\(\displaystyle 180^{\circ}\)
\(\displaystyle 90^{\circ}\)
\(\displaystyle -180^{\circ}\)
\(\displaystyle -90^{\circ}\)
(c)
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
\(\displaystyle 30^{\circ}\)
\(\displaystyle -150^{\circ}\)
\(\displaystyle -210^{\circ}\)
\(\displaystyle 210^{\circ}\)
(d) Draw an angle of measure
\(-225^{\circ} \) in standard position.
Answer .
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Definition 6.1.6 .
A
central angle is an angle whose vertex is at the center of a circle.
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Definition 6.1.7 .
One
radian is the measure of a central angle of a circle that intersects an arc the same length as the radius.
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Activity 6.1.9 .
We now know that one turn around the circle measures
\(360^{\circ}\) and also
\(2\pi\) radians. Use this information to set up a proportion to find the equivalent radian measure of the following angles that are given in degrees.
(a)
\(180^{\circ}\)
\(\displaystyle \frac{\pi}{4}\)
\(\displaystyle \frac{3\pi}{4}\)
\(\displaystyle \frac{\pi}{2}\)
Hint .
Try setting up a proportion!
\(\dfrac{180^\circ}{360^\circ}= \dfrac{x}{2\pi}\)
Answer .
(b)
\(45^{\circ}\)
\(\displaystyle \frac{\pi}{4}\)
\(\displaystyle \frac{3\pi}{4}\)
\(\displaystyle \frac{\pi}{2}\)
Activity 6.1.10 .
Continue using the fact that one turn around the circle measures
\(360^{\circ}\) and also
\(2\pi\) radians. Use this information to set up a proportion to find the equivalent degree measure of the following angles that are given in radians.
(a)
\(\dfrac{\pi}{2}\)
\(\displaystyle 45^{\circ}\)
\(\displaystyle 90^{\circ}\)
\(\displaystyle 180^{\circ}\)
\(\displaystyle 360^{\circ}\)
Hint .
Try setting up a proportion!
\(\dfrac{x}{360^\circ}= \dfrac{\frac{\pi}{2}}{2\pi}\)
Answer .
(b)
\(\frac{3\pi}{4}\)
\(\displaystyle 45^{\circ}\)
\(\displaystyle 90^{\circ}\)
\(\displaystyle 135^{\circ}\)
\(\displaystyle 180^{\circ}\)
Activity 6.1.11 .
We’ll now use the proportions from before to come up with a way to convert between degrees and radians for any given angle. We’ll call \(a\) the angle’s measure in degrees and \(b\) the angle’s measure in radians. So, we have the following proportion that must hold:
\begin{equation*}
\dfrac{a}{360^\circ}= \dfrac{b}{2\pi}
\end{equation*}
(a) Let’s say we know an angle’s measure in degrees,
\(a\text{,}\) and need to find the angle’s measure in radians,
\(b\text{.}\) Solve for
\(b\) in the proportion.
Answer .
\(b=\dfrac{\pi}{180^\circ}\cdot a\)
(b) Use the formula you just developed to convert
\(60^\circ\) to radians. Leave your answer in terms of
\(\pi\text{.}\) Do not approximate!
(c) Now let’s assume we know an angle’s measure in radians,
\(b\text{,}\) and need to find the angle’s measure in degrees,
\(a\text{.}\) Solve for
\(a\) in the proportion.
Answer .
\(a=\dfrac{180^\circ}{\pi}\cdot b\)
(d) Use the formula you just developed to convert
\(\dfrac{\pi}{6}\) to degrees.
Activity 6.1.13 .
Convert each of the following angles.
(a)
\(\dfrac{2\pi}{3}\) radians to degrees
(b)
\(\dfrac{7\pi}{6}\) radians to degrees
(c)
\(240^\circ \) to radians
(d)
\(315^\circ \) to radians
Subsection 6.1.2 Exercises
