CHAPTER 3
LESSON 2
Teacher’s Guide
Angles in Standard Position
AW 3.3
MP 4.1
Objective:
 To define standard position and to investigate the properties of angles in standard
position
1) Angles as Rotations
Any angle may be viewed as the rotation of a ray about its endpoint.
Consider a ray OB that is then rotated about its endpoint O to form AOP with vertex O.
By agreement, we say that such a counter clockwise rotation produces
a positive
angle. In this case, AOP  40.
Side OP is called the initial
Side OP is called the terminal

side or arm of AOP.
side (or arm) of AOP.
Rotating the ray OA in a clockwise direction produces a negative
angle.
In this case, AOP1  40.
As before, side OA is called the initial
side (or arm) of AOP1.
Side OP1 is called the terminal
side (or arm) of AOP1.
2) Definition of Standard Position
To study angles with respect to the Cartesian coordinate system, we must first agree on a
set way to place an angle in the coordinate plane. We say that an angle  is in standard
position in the coordinate plane if the following two requirements are satisfied.
a) The vertex of the angle  is at the origin.
b) The initial side of  lies on the positive x–axis.
Let’s place the angle  below in standard position.


Example 1:
Draw a positive angle  in standard position whose terminal arm is in quadrant 4.
Example 2:
Draw a negative angle  in standard position whose terminal side is in quadrant 2.
Example 3:
3
5
Using the points provided on the grid below as a guide, sketch the angles 4 and  4
in standard position.

3) Coterminal Angles
Definition: Coterminal angles are standard position angles that share a common
_______terminal_arm________________.
The two angles that you drew in Example 3 are coterminal angles.
Every angle in standard position has an infinite number of coterminal angles associated
3
with it. For example, reconsider the standard position angle   4 (135) from
Example 3. Let P(x,y) be a point on the terminal side of .
Let’s move the terminal arm of  counter clockwise one complete rotation about the
3
3
11
origin. Thus the new coterminal angle of 4 will be 4  2  4 . (In degrees, this
new coterminal angle will be 135  360  495. )
3
Let’s move P another full rotation. The new coterminal angle of 4 will thus be
3
19
4  2  2  4 .
In both cases, P will still have the same coordinates (x, y).
3
In the same manner, the terminal arm of 4 can rotate clockwise about the origin.
3
Let’s move the terminal arm of 4 clockwise one complete rotation about the origin.
3
5
Thus the new coterminal angle will be 4  2   4 .
(In degrees, this new coterminal angle will be 135  360  225. )

3
We could also rotate the terminal arm of 4 clockwise one more complete rotation about
3
13
the origin. Thus, another coterminal angle of  is 4  2 (2 )  4 .
In both cases, the point P will still have the same coordinates (x, y).
Of course, we are not limited to the number of times we can rotate the terminal arm of  ,
either clockwise or counter clockwise. Such rotations produce an infinite family of
3
coterminal angles of 4 .
Rotating counter clockwise, we generate the following coterminal angles.



3
3
19

 2  2 
 (2 )  2 
(2 rotations)

4
4
4
3
 Coterminal angles of 4
3  2  2  2  3  (2 )  3  27 (3 rotations)

4
4
4


3  2  2  2  2  3  (2 ) 4  35 (4 rotations)

4
4
4

3  2  11 (1 rotation)
4
4
3
Therefore, k complete counter clockwise rotations of the terminal arm of 4 would
3
produce a coterminal angle of 4  (2 )k radians.
.
Likewise, rotating clockwise, we generate the following negative coterminal angles.



3
3
13

 2  2 
 (2 ) 2  
(2 rotations)

4
4
4

3
 Coterminal angles of 4
3  2  2  2  3  (2 )  3   21 (3 rotations)

4
4
4


3  2  2  2  2  3  (2)  4   29 (4 rotations)

4
4
4

3  2   5  (1 rotation)
4
4
3
Thus, k complete clockwise rotations of the terminal arm of 4 would produce a negative
3
coterminal
.
angle of 4  (2 )k radians.

Generalization:
In general, consider an angle  in standard position. Coterminal angles of  will have
the form   2n radians where n is any integer.
(If  is in degrees, coterminal angles of  will have the form   360n
where n is any integer.)

  2(1)
  2(2)


  2 (1)
   2 (1)

  2 (2)
   2 (2)

etc.
etc.
4) Reference Angles
Definition: Consider an angle  in standard position. The reference angle of  is
the acute angle formed by the terminal side of  , and the x–axis.
Exercise 1:
Sketch the reference angle of each angle  .
1)
2)
3)

4)
Exercise 2:
Consider each of the following angles.
(a)   45
(b)   5
6
(e)   240
(f)  = 
2
3
(c)   225
(d)   5
3
(g)   360
(h)   7
4
(i) For each angle, draw the standard position angle.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(ii) State which quadrant each angle is in.
(a) Quadrant: I
(b) Quadrant: II
(c) Quadrant: III
(d) Quadrant: IV
(e) Quadrant: II
(f) Quadrant: III
(g) Quadrant: I
(h) Quadrant: IV
(iii) For each angle, find two coterminal angles.
(Answers will vary.)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(iv) Find the reference angle of each angle.



6
(a) 45º
(b)
(e) 60

(f) 3
(c) 45º
(d)

3
(g) 0
(h)

4