MCR 3UI ­ U5 ­ D5 ­ Trig Ratios of Any Angle and CAST Rule ­ complete 3.notebook
January 28, 2015
Unit 5: Trigonometry
Day 5: Trig Ratios of Any Angle &
CAST Rule
Today we will.....
1. Learn how to use a terminal arm to draw any angle on a
cartesian plane.
2. Learn how to determine angle measurements given
coordinate points.
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MCR 3UI ­ U5 ­ D5 ­ Trig Ratios of Any Angle and CAST Rule ­ complete 3.notebook
January 28, 2015
Angles in Standard Position
Definitions:
y
Terminal Arm
(rotates around the origin)
6
5
Principal Angle
(angle created by the two arms)
4
3
2
θ
1
­6
­5
­4
­3
­2
­1
0
­1
­2
­3
­4
­5
1
2
x
3
4
5
6
Initial Arm
(always on the x-axis when
in 'standard position')
­6
2
MCR 3UI ­ U5 ­ D5 ­ Trig Ratios of Any Angle and CAST Rule ­ complete 3.notebook
January 28, 2015
CAST Rule
Given angle θ formed by a terminal arm in standard position:
If our Point was (x, y):
"How do we determine the length of r?"
y
6
5
4
-always positive
Point(x,y)
3
r
2
1
­6
­5
­4
­3
­2
­1
θ
0
­1
1
x
2
3
4
5
6
­2
Note: All 3 ratios are
positive.
­3
­4
­5
­6
If our Point was (-x, y):
y
6
5
4
Point (-x,y)
3
r
2
1
θ
­6
­5
­4
­3
­2
­1
0
­1
x
1
2
3
4
5
6
Note: Sine is the only
positive ratio in this
quadrant.
­2
­3
­4
­5
­6
3
MCR 3UI ­ U5 ­ D5 ­ Trig Ratios of Any Angle and CAST Rule ­ complete 3.notebook
January 28, 2015
If our Point was (-x, -y):
y
6
5
4
3
2
1
­6
­5
­4
­3
θ
­2
­1
x
0
­1
1
2
3
4
5
6
­2
r
Note: Tan is the only
positive ratio in this
quadrant.
­3
Point(-x, -y)
­4
­5
­6
If our Point was (x, -y):
y
6
5
4
3
2
1
­6
­5
­4
­3
­2
­1
0
­1
x
1
2
θ
3
4
5
6
Note: Cos is the only positive
ratio in this quadrant.
­2
­3
r
­4
­5
­6
Point(x, -y)
Note: θ is measured from the origin, rotating counter clockwise.
However, to use simple trig ratios, we need a Right-Triangle,
hence, θ is always placed between the x-axis and the terminal
arm.
(So we can make a Right Triangle with the arm and the x-axis)
4
MCR 3UI ­ U5 ­ D5 ­ Trig Ratios of Any Angle and CAST Rule ­ complete 3.notebook
January 28, 2015
So, the CAST rule: "Did you figure it out yet?"
To determine whether a trig ratio will be positive or negative, the
CAST rule was made: If you know which quadrant θ lies in, then:
y
6
Sin only +ve
II
5
4
3
All ratios + ve
I
2
1
­6
­5
­4
­3
­2
­1
Tan only +ve
x
0
­1
1
2
3
4
5
­2
Cos only +ve
6
­3
III
­4
IV
­5
­6
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MCR 3UI ­ U5 ­ D5 ­ Trig Ratios of Any Angle and CAST Rule ­ complete 3.notebook
January 28, 2015
Example 1: The point P(-3, -6) lies on the terminal arm of an
angle θ in standard position.
a) Determine the exact values of sin θ, cos θ and tan θ.
b) Determine the principal angle, θ.
Hint to find θ
To find
use th
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MCR 3UI ­ U5 ­ D5 ­ Trig Ratios of Any Angle and CAST Rule ­ complete 3.notebook
January 28, 2015
Example 2: The point P(10, -5) lies on the terminal arm of an
angle θ in standard position.
a) Determine the exact values of sin θ, cos θ and tan θ.
b) Determine the principal angle, θ.
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MCR 3UI ­ U5 ­ D5 ­ Trig Ratios of Any Angle and CAST Rule ­ complete 3.notebook
January 28, 2015
Example 3: Angle θ is in standard position in
quadrant II and 0o ≤ θ ≤ 360o. Given the trig
ratio, find:
a) the exact values of the other two trig ratios.
b) the principal angle, θ
8
MCR 3UI ­ U5 ­ D5 ­ Trig Ratios of Any Angle and CAST Rule ­ complete 3.notebook
January 28, 2015
Today's Practice Questions:
pg. 348 #1abef + principal angle for all
#2abef + principal angle for all
#6
Quiz in 2 Classes:
Test in 4 Classes
Note:
When the textbook says 0 ≤ θ ≤ 2π ,
they mean: 0o ≤ θ ≤ 360o
Correction: 1a) cosθ = 8
17
9