Title | Lesson 3 - Notes from class |
---|---|
Course | Trigonometry |
Institution | University of Toledo |
Pages | 24 |
File Size | 739 KB |
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Notes from class ...
LESSON 3 REFERENCE ANGLES Topics in this lesson: 1. THE DEFINITION AND EXAMPLES OF REFERENCE ANGLES 2. THE REFERENCE ANGLE OF THE SPECIAL ANGLES 3. USING A REFERENCE ANGLE TO FIND THE VALUE OF THE SIX TRIGONOMETRIC FUNCTIONS 1.
THE DEFINITION AND EXAMPLES OF REFERENCE ANGLES
Definition The reference angle of the angle , denoted by ' , is the acute angle determined by the terminal side of and either the positive or negative x-axis. Recall, an acute angle is an angle whose measurement is greater than 0 and less than 90 . NOTE: By definition, the reference angle is an acute angle. Thus, any angle whose terminal side lies on either the x-axis or the y-axis does not have a reference angle. Examples Find the reference angle ' for the following angles made by rotating counterclockwise. 1.
is in the I quadrant.
Animation of the reference angle.
'
Relationship between and ' : ' Toledo
2.
is in the II quadrant.
Animation of the reference angle.
'
Relationship between and ' : ' OR ' 180 OR ' OR ' 180 3.
is in the III quadrant.
Animation of the reference angle.
' Relationship between and ' : ' OR ' 180 OR ' OR 180 ' Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
4.
is in the IV quadrant.
Animation of the reference angle.
' Relationship between and ' : ' 2 OR ' 360 OR ' 2 OR ' 360 Examples Find the reference angle ' for the following angles made by rotating clockwise. 1.
is in the IV quadrant.
Animation of the reference angle.
' Relationship between and ' : ' oledo
2.
is in the III quadrant.
Animation of the reference angle.
'
Relationship between and ' :
'
OR
' 180
OR
' OR 3.
' 180
is in the II quadrant.
Animation of the reference angle.
'
Relationship between and ' : ' OR ' 180 OR ' Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
OR
180 '
4.
is in the I quadrant.
Animation of the reference angle.
'
Relationship between and ' : ' 2 OR ' 360 OR
' 2
OR
' 360
Examples Find the reference angle for the following angles. 1.
140 The angle is in the II quadrant.
Animation of the reference angle.
'
' = 180 140 40 do
2.
350 The angle is in the I quadrant.
Animation of the reference angle.
'
' = 360 350 10 3.
9 7
The angle is in the III quadrant.
Animation of the reference angle.
' ' =
9 9 7 2 = = 7 7 7 7 edo
4.
9 7
The angle is in the II quadrant.
Animation of the reference angle.
'
' =
5.
9 7 2 9 = = 7 7 7 7
7 15
The angle is in the IV quadrant.
Animation of the reference angle.
'
' f Toledo
7 15
6.
79 42
The angle is in the IV quadrant.
Animation of the reference angle.
'
' = 2
7.
79 84 79 5 = = 42 42 42 42
150
Animation of the reference angle.
'
' = 180 150 30 do
8.
180 The angle is on the negative x-axis. Thus, the angle does not have a reference angle.
9.
3 2
The angle is on the positive y-axis. Thus, the angle does not have a reference angle. Back to Topics List
2.
THE REFERENCE ANGLE OF THE SPECIAL ANGLES
The reference angle of the Special Angles of
5 7 11 , , , and is . 6 6 6 6 6
The reference angle of the Special Angles of
7 3 5 , , , and is . 4 4 4 4 4
The reference angle of the Special Angles of
5 2 4 , , , and is . 3 3 3 3 3
The reference angle of the Special Angles of 330 is 30 .
30 , 150 , 210 , and
The reference angle of the Special Angles of 45 , 135 , 225 , and 315 is 45 . The reference angle of the Special Angles of 300 is 60 .
60 , 120 , 240 , and
Back to Topics List Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
3.
USING A REFERENCE ANGLE TO FIND THE VALUE OF THE SIX TRIGONOMETRIC FUNCTIONS
To find the value of a trigonometric function of an angle in the II, III, and IV quadrants by rotating counterclockwise, we will make use of the reference angle of the angle. To find the value of a trigonometric function of an angle in the I, II, III, and IV quadrants by rotating clockwise, we will make use of the reference angle of the angle. Theorem Let ' be the reference angle of the angle . Then 1.
cos cos '
4.
sec sec '
2.
sin sin '
5.
csc csc '
3.
tan tan '
6.
cot cot '
where the sign of + or is determined by the quadrant that the angle is in.
We will use this theorem to help us find the exact value of the trigonometric functions of angles whose terminal side lies in the first (rotating clockwise), second, third, and fourth quadrants. We do not need any help to find the exact value of the trigonometric functions of angles whose terminal side lies on one of the coordinate axes. We know the x-coordinate and the y-coordinate of the point of intersection of these angles with the Unit Circle. So, we know the exact value of the cosine, sine, and tangent of these angles from the x-coordinate, y-coordinate, and the y-coordinate divided by the x-coordinate, respectively. This might be the reason that a reference angle is not defined for an angle whose terminal side lies on one of the coordinate axes.
The cosine, sine, and tangent of the special angles with a denominator of 6 in radian angle measurement. The cosine, sine, and tangent of the special angles with a denominator of 4 in radian angle measurement. The cosine, sine, and tangent of the special angles with a denominator of 3 in radian angle measurement. Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
Examples Use a reference angle to find the exact value of the six trigonometric functions of the following angles. 1.
2 3
(This is the 12 0 angle in units of degrees.)
2 is in the II quadrant. The reference angle of the angle 3 2 is the angle ' . 3 3
The angle
2 is in the II quadrant, then the 3 2 2 point of intersection P 3 of the terminal side of the angle with 3 the Unit Circle is in the II quadrant. Thus, the x-coordinate of the point 2 P is negative and the y-coordinate of this point is positive. Since 3 2 2 2 cos , then cos is the x-coordinate of the point P is negative. 3 3 3 2 2 2 is the y-coordinate of the point P 3 , then sin is Since sin 3 3 2 2 divided by positive. Since tan is the y-coordinate of the point P 3 3 2 is negative since a positive the x-coordinate of this point, then tan 3 divided by a negative is negative. Since the terminal side of the angle
cos
2 1 cos 3 3 2
sec
2 2 3
sin
3 2 sin 3 3 2
csc
2 2 3 3
Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
tan
2.
210
2 tan 3 3
cot
3
(This is the
2 1 3 3
7 angle in units of radians.) 6
The angle 210 is in the III quadrant. The reference angle of the angle 210 is the angle ' 30 . Since the terminal side of the angle 210 is in the III quadrant, then the point of intersection P ( 210 ) of the terminal side of the angle 210 with the Unit Circle is in the III quadrant. Thus, the x-coordinate of the point P ( 210 ) is negative and the y-coordinate of this point is negative. Since cos 210 is the x-coordinate of the point P ( 210 ) , then cos 210 is negative. Since sin 210 is the y-coordinate of the point P ( 210 ) , then sin 210 is negative. Since tan 210 is the y-coordinate of the point P ( 210 ) divided by the x-coordinate of this point, then tan 210 is positive since a negative divided by a negative is positive.
3
cos 210 cos 30
sin 210 sin 30
tan 210 tan 30
3.
7 4
2
1 2
1 3
sec 210
csc 210 2
cot 210
(This is the 315 angle in units of degrees.)
Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
2 3
3
7 is in the IV quadrant. The reference angle of the angle 4 7 is the angle ' . 4 4
The angle
7 is in the IV quadrant, then the 4 7 7 point of intersection P 4 of the terminal side of the angle with 4 the Unit Circle is in the IV quadrant. Thus, the x-coordinate of the point 7 P is positive and the y-coordinate of this point is negative. Since 4 7 7 7 cos is the x-coordinate of the point P 4 , then cos is positive. 4 4 7 7 7 , then sin is the y-coordinate of the point P is Since sin 4 4 4 7 7 is the y-coordinate of the point P 4 divided by negative. Since tan 4 7 tan the x-coordinate of this point, then is negative since a negative 4 divided by a positive is negative. Since the terminal side of the angle
sec
7 2 4 2
2 7 sin 4 4 2
csc
7 2 4 2
7 tan 1 4 4
cot
7 1 4
cos
7 cos 4 4
sin
tan
2 2
Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
2
2
4.
30
angle in units of radians.) 6
(This is the
The angle 30 is in the IV quadrant. The reference angle of the angle 30 is the angle ' 30 . Since the terminal side of the angle 30 is in the IV quadrant, then the point of intersection P ( 30 ) of the terminal side of the angle 30 with the Unit Circle is in the IV quadrant. Thus, the x-coordinate of the point P ( 30 ) is positive and the y-coordinate of this point is negative. Since cos ( 30 ) is the x-coordinate of the point P ( 30 ) , then cos ( 30 ) is positive. Since sin ( 30 ) is the y-coordinate of the point P ( 30 ) , then sin ( 30 ) is negative. Since tan ( 30 ) is the y-coordinate of the point P ( 30 ) divided by the x-coordinate of this point, then tan ( 30 ) is negative since a negative divided by a positive is negative.
cos ( 30 ) cos 30
3 2
sin ( 30 ) sin 30
tan ( 30 ) tan 30
5.
5 6
sec ( 30 )
1 2
1 3
2 3
csc ( 30 ) 2
cot ( 30 ) 3
(This is the 15 0 angle in units of degrees.)
5 is in the III quadrant. The reference angle of the angle 6 5 is the angle ' . 6 6
The angle
Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
5 is in the III quadrant, then the 6 5 5 of the terminal side of the angle point of intersection P 6 6 with the Unit Circle is in the III quadrant. Thus, the x-coordinate of the point 5 P is negative and the y-coordinate of this point is negative. Since 6 5 5 5 cos , then cos is the x-coordinate of the point P is 6 6 6 5 5 , then negative. Since sin 6 is the y-coordinate of the point P 6 5 5 sin is negative. Since tan is the y-coordinate of the point 6 6 5 5 P divided by the x-coordinate of this point, then tan is 6 6 positive since a negative divided by a negative is positive. Since the terminal side of the angle
6.
3 5 cos cos 6 2 6
2 5 sec 3 6
1 5 sin sin 6 2 6
5 csc 2 6
1 5 tan tan 6 3 6
5 cot 6
225
(This is the
3
5 angle in units of radians.) 4
The angle 225 is in the II quadrant. The reference angle of the angle 225 is the angle ' 45 . Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
Since the terminal side of the angle 225 is in the II quadrant, then the point of intersection P ( 225 ) of the terminal side of the angle 225 with the Unit Circle is in the II quadrant. Thus, the x-coordinate of the point P ( 225 ) is negative and the y-coordinate of this point is positive. Since cos ( 225 ) is the x-coordinate of the point P ( 225 ) , then cos ( 225 ) is negative. Since sin ( 225 ) is the y-coordinate of the point P ( 225 ) , then sin ( 225 ) is positive. Since tan ( 225 ) is the y-coordinate of the point P ( 225 ) divided by the x-coordinate of this point, then tan ( 225 ) is negative since a positive divided by a negative is negative.
cos ( 225 ) cos 45
sin ( 225 ) sin 45
2 2
2 2
tan ( 225 ) tan 45 1
7.
5 3
2 2
sec ( 225 )
csc ( 225 )
2 2
2
2
cot ( 225 ) 1
(This is the 30 0 angle in units of degrees.)
5 is in the I quadrant. The reference angle of the angle 3 5 is the angle ' . 3 3
The angle
5 is in the I quadrant, then the 3 5 5 of the terminal side of the angle point of intersection P 3 3 with the Unit Circle is in the I quadrant. Thus, the x-coordinate of the point Since the terminal side of the angle
Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
5 P is positive and the y-coordinate of this point is positive. Since 3 5 5 5 cos , then cos is the x-coordinate of the point P is 3 3 3 5 5 sin , then P positive. Since is the y-coordinate of the point 3 3 5 5 sin is positive. Since tan is the y-coordinate of the point 3 3 5 5 P divided by the x-coordinate of this point, then tan is 3 3 positive since a positive divided by a positive is positive.
8.
1 5 cos cos 3 2 3
5 sec 2 3
3 5 sin sin 3 2 3
2 5 csc 3 3
5 tan tan 3 3
1 5 cot 3 3
240
3
(This is the
4 angle in units of radians.) 3
The angle 240 is in the III quadrant. The reference angle of the angle 240 is the angle ' 60 .
cos 240 cos 60
1 2
Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
sec 240 2
sin 240 sin 60
tan 240 tan 60
9.
5 6
3
csc 240
2
cot 240
3
2 3
1 3
(This is the 15 0 angle in units of degrees.)
5 is in the II quadrant. The reference angle of the angle 6 5 is the angle ' . 6 6
The angle
10.
cos
3 5 cos 6 6 2
sec
5 2 6 3
sin
5 1 sin 6 6 2
csc
5 2 6
tan
5 1 tan 6 6 3
cot
5 3 6
45
(This is the
angle in units of radians.) 4
The angle 45 is in the IV quadrant. The reference angle of the angle 45 is the angle ' 45 .
cos ( 45 ) cos 45
2 2
sec ( 45 )
Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
2 2
2
sin ( 45 ) sin 45
2
csc ( 45 )
2
tan ( 45 ) tan 45 1
11.
330
2 2
2
cot ( 45 ) 1
(This is the
11 angle in units of radians.) 6
The angle 330 is in the I quadrant. The reference angle of the angle 330 is the angle ' 30 .
3 2
cos ( 330 ) cos 30
sin ( 330 ) sin 30
tan ( 330 ) tan 30
sec ( 330 )
3
1 2
csc ( 330 ) 2
1
cot ( 330 )
3
12.
5 4
2
3
(This is the 225 angle in units of degrees.)
5 is in the III quadrant. The reference angle of the angle 4 5 is the angle ' . 4 4
The angle
cos
2 5 cos 4 4 2
sec
Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330
5 2 4 2
2
13.
...