Lesson 3: Exploring Trigonometric Ratios for Angles
Greater than 90°
Part A – Definitions
• An angle is in standard position if its vertex is at the origin, (0, 0), in the Cartesian Plane,
and one fixed arm (the initial arm) is placed along the positive x-axis. The terminal arm is
the final position of the rotating arm. Angle θ is measured from the initial arm to the
terminal arm.
y
terminal arm
amount of rotation
θ
x
initial arm
Origin
 Principal Angle: The counterclockwise angle between the initial arm and the terminal
arm in standard position. Its value is between 0 and 360 .
 Negative Angle : An angle measured clockwise from the positive x-axis.
y
Note: Positive rotation is
counterclockwise
y
θ
x
x
θ
Positive Rotation
(counterclockwise)
Negative Rotation
(clockwise)
• Related acute angle is the angle (between 0º and 90º) between the closest x-axis
and the
terminal arm of an angle in standard position when the terminal arm lies in
first, second
or third quadrants. (i.e. the angle relative to the x-axis.)
• The quadrants of a rectangular coordinate system are as follows:
terminal arm
y
quadrant 2
quadrant 1
θ
Relative acute angle
Principal angle
β
x
Vertex
quadrant 3
initial arm
quadrant 4
2
1
3
4
Example:
Draw a principal angle that is 135o. Show it’s related acute angle.
Trig ratios in standard position can also be expressed in terms of x, y and r.
r
y
x
Let’s look at our special 45o triangle:
sin 45o =
cos 45o =
tan 45o =
45o
Which ratio(s) is (are) positive?
Let’s move the triangle to Quadrant 2:
What is the principal angle?
What is the related acute angle?
sin 45o =
cos 45o =
45o
Which ratio(s) is (are) positive?
tan 45o =
… in Quadrant 3:
What is the principal angle?
What is the related acute angle?
45o
sin 45o =
cos 45o =
tan 45o =
Which ratio(s) is (are) positive?
… in Quadrant 4:
What is the principal angle?
What is the related acute angle?
45o
sin 45o =
cos 45o =
Which ratio(s) is (are) positive?
The CAST Rule:
Sin
All
Tan
Cos
The cast rule helps you remember which trig ratios are positive in each quadrant.
tan 45o =
Using the appropriate special triangle, sketch each angle and determine the trig ratio:
a) cos 60o
d)
cos 120o
b)
tan 30o
c)
sin 45o
e)
sin 210o
f)
cot 315o