MCT4C
Rotation Angles and the CAST Rule
Unit 5 Lesson 8
A rotation angle is one that can be plotted using Cartesian co-ordinates. We move trigonometry onto the
Cartesian plane because this vastly expands our ability to apply this branch of mathematics. In a triangle,
180o . When an angle is rotated on the
angle measures are limited to values between 0 o and 180 o , 0o
Cartesian plane, we can represent both positive and negative angles, and the measure of the angle is
unlimited.
As illustrated below, a rotation angle is defined in terms of an initial arm, lying on the positive x-axis, and a
terminal arm which rotates about the vertex at the origin.
If the terminal arm rotates counter-clockwise,
the angle is positive.
y
If the terminal arm rotates clockwise,
the angle is negative.
y
Pa
Pb
a
x
b
x
Note in the examples above, that different rotation angles may have the same terminal arm. a has rotated
through 150 o , while b has rotated through 210 o . However, the terminal arm is in the same position.
360 o 210 o would also share this same terminal arm.
The angles 510 o 360 0 150 0 , and 570 o
Angles that share the same terminal arm are known as co-terminal.
The principal angle ( A) is the smallest positive angle relative to the positive x-axis that is created by the
terminal arm, 0 o A 360 o . Every principal angle has a related acute angle which is found in a triangle
created by drawing a line starting at point P (at the end of the terminal arm) and then travelling up or down to
the x-axis such that it is perpendicular to that axis.
For principal a above, the related acute angle is found in quadrant 2 and has a measure of 30 o .
a 1500 180o 30o .
We think:
b is not a principal angle as it has a negative measure. We still use the measure of the related acute angle
b
210o
180o 30o .
to determine the measure of b .
We think:
When the right triangle illustrating any principal angle
is drawn, that triangle will always be in one of the four
positions shown to the right. The length of the terminal
arm is referred to as r, as the rotating terminal arm makes
a circle with radius length r.
P2
y
P1
2
3
P3
x
1
4
P4
Page 2
U5/L8
Finding the primary trigonometric ratios given the end point of the terminal arm.
Eg 1)
A is formed by a terminal arm that rotates to point P 4, 9 . Find:
a) the primary trigonometric ratios for A .
b) the measure of the related acute angle .
(note that some ratios are negative)
(use the positive primary ratio to calculate this)
c) the principal measure of A .
d) What are two possible measures for angles
co-terminal to A ?
Eg. 2) B is formed by a terminal arm that rotates to point P
3, 1 . Find:
a) the primary trigonometric ratios for B .
b) the measure of the related acute angle
c) the principal measure of
.
B.
d) What are two possible measures for angles
co-terminal to B ?
Page 3
U5/L8
The CAST Rule
It was evident in examples 1 and 2 that the sign of the trigonometric ratios varies with the quadrant. The
acronym CAST enables us to identify the trig ratios having positive values in any quadrant. The sign of the
ratio is based upon the sign of the x and y co-ordinates in the particular quadrant.
y
S
C
T
(2)
x, y
S
C
T
S
C
T
(1)
x, y
x
S
C
T
(3)
x,
y
x,
y
(4)
You may have noticed in lessons 5 and 6 that, because of the CAST rule:
1)
the sine ratios for any two supplementary angles are positive AND equal in magnitude.
Using your calculator, try sin 55o and sin 125 o .
2)
the cosine ratios for any two supplementary angle are equal in magnitude but have opposite
signs.
Using your calculator, try cos15 o and cos165 0 .
Homework
Extra answers:
text p 282 #1a-e, 2bcdh,3, 4, 6ace, 8abc, 12, 18a-d, 19
12. extra – find the measure of the primary angle and the measures of
A, B, C , and D
12.
53 o ,
A
53o ,
B 127 o ,
C
233o ,
D
307 o