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
angle. In this case, ∠AOP = +40°.
a positive
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 θ is in standard
position in the coordinate plane if the following two requirements are satisfied.
a) The vertex of the angle θ is at the origin.
b) The initial side of θ lies on the positive x–axis.
Let’s place the angle θ below in standard position.
Example 1:
Draw a positive angle θ in standard position whose terminal arm is in quadrant 4.
Example 2:
Draw a negative angle θ in standard position whose terminal side is in quadrant 2.
Example 3:
5π
Using the points provided on the grid below as a guide, sketch the angles 3π
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
°
with it. For example, reconsider the standard position angle θ = 3π
4 (135 ) from
Example 3. Let P(x,y) be a point on the terminal side of θ.
Let’s move the terminal arm of θ counter clockwise one complete rotation about the
3π + 2π = 11π .
origin. Thus the new coterminal angle of 3π
4 (In degrees, this
4 will be 4
new coterminal angle will be 135° + 360° = 495°.)
Let’s move P another full rotation. The new coterminal angle of 3π
4 will thus be
3π + 2π + 2π = 19π .
4
4
In both cases, P will still have the same coordinates (x, y).
In the same manner, the terminal arm of 3π
4 can rotate clockwise about the origin.
Let’s move the terminal arm of 3π
4 clockwise one complete rotation about the origin.
3π
5π
Thus the new coterminal angle will be 4 − 2π = − 4 .
(In degrees, this new coterminal angle will be 135° − 360° = −225°. )
We could also rotate the terminal arm of 3π
4 clockwise one more complete rotation about
3π
−13π
the origin. Thus, another coterminal angle of θ is 4 − 2⋅ (2π ) = 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 θ ,
either clockwise or counter clockwise. Such rotations produce an infinite family of
coterminal angles of 3π
4 .
Rotating counter clockwise, we generate the following coterminal angles.



3π
3π
19π

+ 2π + 2π =
+ (2π ) ⋅ 2 =
(2 rotations)

4
4
4
3π
 Coterminal angles of 4
3π
3π

+ 2π + 2π + 2π =
+ (2π ) ⋅ 3 = 27π (3 rotations)
4
4

4

3π
35π
3π
+ 2π + 2π + 2π + 2π =
+ (2π )⋅ 4 =
(4 rotations)
4
4
4

11π
3π
+ 2π =
(1 rotation)
4
4
Therefore, k complete counter clockwise rotations of the terminal arm of 3π
4 would
3π
produce a coterminal angle of 4 + (2π )k radians.
.
Likewise, rotating clockwise, we generate the following negative coterminal angles.



3π
3π
13π

− 2π − 2π =
− (2π )⋅ 2 = −
(2 rotations)

4
4
4
3π
 Coterminal angles of 4
3π
3π

− 2π − 2π − 2π =
− (2π ) ⋅ 3 = − 21π (3 rotations)
4
4

4

3π
29π
3π
− 2π − 2π − 2π − 2π =
− (2π) ⋅ 4 = −
(4 rotations)
4
4
4

π
3π
− 2π = − 5 (1 rotation)
4
4
Thus, k complete clockwise rotations of the terminal arm of 3π
4 would produce a negative
3π
coterminal angle of 4 − (2π )k radians.
.
Generalization:
In general, consider an angle θ in standard position. Coterminal angles of θ will have
the form θ + 2π n radians where n is any integer.
(If θ is in degrees, coterminal angles of θ will have the form θ + 360n°
where n is any integer.)
θ
θ − 2π (1)
= θ + 2π (−1)
θ + 2π(1)
θ − 2π (2)
= θ + 2π (−2)
θ + 2π(2)
etc.
etc.
4) Reference Angles
Definition: Consider an angle θ in standard position. The reference angle of θ is
the acute angle formed by the terminal side of θ , and the x–axis.
Exercise 1:
Sketch the reference angle of each angle θ .
1)
2)
3)
4)
Exercise 2:
Consider each of the following angles.
(a) θ = 45°
(b) θ = 5π
6
(e) θ = − 240°
(f) θ = −
2π
3
(c) θ = −225°
(d) θ = 53π
(g) θ = 360°
(h) θ = 7 π
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.
π
6
(a) 45º
(b)
(e) 60°
(f) π
3
π
3
(c) 45º
(d)
(g) 0°
(h) π
4