Download 9.2 Angles of Rotation

9.2 Angles of Rotation
Term
Initial side
Definition
Where the rotation angle
starts
The ray on the positive xaxis
terminal
side
Also referred to as 0
Terminal side
Positive
angles
Negative
angles
Coterminal
angles
Angle of
rotation
Standard
position
y
135
x
initial
side
-225
Where the rotation angle
stops
Angles that go counterclockwise from 0
Angles that go clockwise from 0
Angles that share a terminal side, or stop at the same
place.
Formed by rotating the terminal side and keeping the
initial side in place
An angle is in standard position when its vertex is at the
origin and one ray is on the positive x-axis.
Starting at 0, and rotating in the positive direction, through which
Quadrants will you pass, and in what order?
Drawing Angles in Standard Position
Draw and label the terminal sides of these rotation angles. Then tell the
Quadrant (or axis) in which the terminal side lies.
Angle
Quad.
Angle
A) 75
F) 315
B) 195
G) 390
C) -120
H) -45
D) -210
I) 630
E) 270
J) -240
Quad.
Finding Coterminal Angles
Which pairs of angles above are coterminal?
What is the degree measure that separates the angles in each pair?
From this, we can develop the following rule for determining what
other rotation angles are coterminal with an angle of  degrees:
If an angle measures , then it is coterminal with angles
measuring (  360n).
Finding Reference Angles
Because rotation angles can be very large, they
are often described using a reference angle,
which is defined as the positive acute angle
formed by the terminal side of θ and the x-axis.
y
For example, a rotation angle of 135 has a 45
reference angle.
B
y
135
45
x
Reference
angle
Determine the reference angle for each
rotation angle below.
A
Rotation
angle
C
y
y
207
320
x
x
x
?
?
?
-150
D
E
y
F
y
508
x
-280
x
y
53
x
Find measures for rotation angles (between -360 and 360) that have
the given characteristics.
1. A positive angle that terminates in Quadrant III with a 71 reference
angle
2. A negative angle that terminates in Quadrant III with a 17 reference
angle
3. A positive angle that terminates in Quadrant II with a 24 reference
angle
4. A positive angle that terminates in Quadrant IV with a 24 reference
angle
Finding Values of Trigonometric Functions in Angles of Rotation
In addition to finding values of
trigonometric functions for
angles in right triangles, we can
also define the same functions
in terms of angles of rotation.
Consider an angle in standard
position, whose terminal side
intersects a circle of radius r.
We can think of the radius as
the hypotenuse of a right
triangle:
The point
where the terminal side of the angle intersects the circle
tells us the lengths of the two legs of the triangle. Now, we can define
the trigonometric functions in terms of
, and :
We can also extend these functions to include non-acute angles.
Example 1: The point (-3, 4) is a
point on the terminal side of an
angle in standard position.
Determine the values of the six
trigonometric functions of the
angle.
Solution:
Notice that the angle is more
than 90 degrees, and that the
terminal side of the angle lies in
the second quadrant. This will
influence the signs of the
trigonometric functions.
Note:
1. The value of r depends on the coordinates of the given point. You
can always find the value of r using the Pythagorean Theorem.
2. The values of the trigonometric functions may be positive or
negative depending upon the quadrant in which the terminal side
of the angle lies. Therefore, it is important to include the signs of
the x- and y-coordinates when determining the value of the
trigonometric functions.
Practice Problems:
1. P(
) is a point on the terminal side of θ in standard position.
Find the exact value of the six trigonometric functions for θ.
sin θ=
cos θ=
tan θ=
csc θ=
sec θ=
cot θ=
2. P(
) is a point on the terminal side of θ in standard position.
Find the exact value of the six trigonometric functions for θ.
sin θ=
cos θ=
tan θ=
csc θ=
sec θ=
cot θ=