LESSON 5-3:
Evaluating Trigonometric Ratios of Angles
Greater than 90o
Placing Angles on the Cartesian X-Y Plane
y
<<
<<
x
<<
<<



A PRINCIPAL angle,  is formed by rotating a ray about a fixed point called the
The ray is called the
at the beginning of the angle and the
at the end of the angle.
An angle,  is in
if :

the vertex of the angle is at the origin,

the initial arm lies fixed along the positive x-axis, and

the terminal arm is the final position of the rotating ray anywhere on the arc of
rotation.
Trig Ratios of Principle Angles
 Rotating a point on the circumference of a circle creates a principle angle, .
 The side opposite  is
and the side adjacent to  is
.
 We choose
as the hypotenuse since it represents the radius of a circle.
y
P(x, y)

x
TRIGONOMETRIC ratios can be defined in terms of the x and y co-ordinates
of any point P(x, y):
If r
2
= x2 + y2 , r =
and r > 0, then:
1.
sin  =
csc  =
2.
cos  =
sec  =
3.
tan  =
cot  =
.
Angle Direction : An angle can have a positive or negative value:
y
y
x
A positive angle is formed by a counter-clockwise rotation of the terminal arm.
A negative angle is formed by a clockwise rotation of the terminal arm.
Coterminal Angles : Angles in standard position that have the same TERMINAL arm.
Ex.1: State two coterminal angles for a) 30 and for b) 120 .
x
Investigation 1: The “CAST” Rule
Complete the table below to find the SIGN of each trig ratio for the angle in each quadrant:

I
in QUADRANT:
II
III
IV
y
r
x
cos  
r
y
tan  
x
II
I
sin  
III
Which ratio is +ve
in this quadrant?
IV
Conclusion: The “CAST” Rule determines which trig ratio is POSITIVE in each quadrant.
Investigation 2: Trig Ratios of Angles Greater than 90˚
1.Complete the table below:
30˚
Angle, 
sin 
cos 
tan
150˚
210˚
330˚
390˚
-30˚
2. Draw the angles above on the Cartesian Plane:
Related acute angle(  ):
Ex. If

= 240°, then
 =
Conclusions:
1. If the principal angle,
.
.
 , is greater than 90 o, then a
,
is formed.
2. The related acute angle,  , and the
angles greater than 90 o
can be used to find trig ratios of
Ex. 1: The point (–6, 8) is on the terminal arm of an angle  in standard position.
a) Determine the primary trig ratios for the principal angle,  .
b) What is the measure of  ?
Ex. 2: Find all possible measures of angle  to the nearest degree if 0    360.
Use CAST.
a) sin  = 0.4226
b) cos  = -
4
5
c) cot  =
Ex. 3: Find the exact value of the sine, cosine, and tangent of  if:
a)  = 240o
b)  = 315o
Ex. 4: Use the point P(0,1) to determine the primary trig ratios for a 90 angle.
Homework: p. 281 #2, 3odds, 5, 6, 10, 11
p. 348 #1-3 (tan ratio only) 5, 6c, 7ce, 11fij,12,13b, 17
 3
Reviewing Trig Ratios of Angles Greater than 90˚
Recall:
Given a point on the terminal arm of any angle in the x-y plane,
P(x,y), we may find the primary trig ratios for the principal angle, 
and the measure of that angle as follows:
1. Create a right triangle between the terminal arm and the closest x-axis.
P(x,y)


2. Label x, y, r and the related acute angle,β, in this right angle triangle.
3. Find the required trig ratio for the related acute angle, β using:
y
x
y
sin  
cos  
tan   .
r
r
x
and then determine the measure of angle  .
4. Find the measure of the principal angle,  by:
(i)
using the CAST Rule to determine # of possible answers
(ii)
add or subtract  to 180 or to 360 to determine the
principal angle  .
Quadrant
I
II
III
IV
Principal Angle Measure, 

180 - 
180 + 
360 - 