Unit 5: Trigonometry
Day 5: Trig Ratios of Any Angle & CAST Rule
Angles in Standard Position
Definitions:
CAST Rule
Given angle θ formed by a terminal arm in standard position:
If our Point was (x, y):
If our Point was (-x, y):
"How do we determine the length of r?"
If our Point was (-x, -y):
If our Point was (x, -y):
Note: θ is measured from the origin, rotating _______________________.
However, to use simple trig ratios, we need a Right-Triangle, hence, θ is always
placed between the ____-axis and the terminal arm.
(So we can make a Right Triangle with the arm and the ____-axis)
So, the CAST rule:
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:
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.
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.
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
𝑠𝑖𝑛𝜃 =
3
4