MCR3U1
U4L4
EVALUATING TRIGONOMETRIC RATIOS
for Any Angle Between 00 and 3600
(Part 1)
PART A ~ INTRODUCTION
The CAST rule helps as a reminder of which trigonometric ratios are
positive in each quadrant:
S A
T C
PART B ~ TRIGONOMETRIC RATIOS for ANGLES BETWEEN 00 & 3600
The trigonometric ratios for any principal angle, , in standard position, where
00    3600, can be determined by finding the related acute angle using coordinates
of any point P(x, y) that lies on the terminal arm of the angle.
Let P(x, y) be a point on the terminal arm of an angle θ in standard position.
y
y
𝑟 = √𝑥 2 + 𝑦 2
P(x, y)
r
P(x, y)
y
r
y

θ
x
x
θ
x
Note: Always drop a  to the x-axis.
x
(  is the related acute angle )
For any angle θ in standard position, with point P(x, y) on the terminal arm, the six
trigonometric ratios are defined as:
sin  
y
r
O
 
H
cos 
x
r
 A
 
H
tan 
y
x
O
 
 A
csc 
r
y
H
 
O
sec 
r
x
H
 
 A
cot 
x
y
 A
 
O
MCR3U1
U4L4
The point P(3, 4) lies on the terminal arm of an angle  in standard position.
Ex 
a)
b)
c)
Sketch the angle.
Determine the exact ratio for sin, cos, and tan.
Repeat the above steps for points (–3, 4), (–3, –4), and (3, –4).
Q2
Sine
 = ____
y
Q1
All
sinθ =
sinθ =
cosθ =
cosθ =
(3,4)
tanθ =
 = ____
tanθ =

x
 = ____
Ex 
sinθ =
sinθ =
cosθ =
cosθ =
tanθ =
tanθ =
 = ____
Tangent
Cosine
Q3
Q4
The point P(5, –12) lies on the terminal arm of angle  in standard position.
a)
b)
Determine the exact values of sin, cos, and tan.
State the value of , using both a counterclockwise & a clockwise rotation.
y
x
MCR3U1
U4L4
SUMMARY of STEPS:
1.
2.
3.
4.
Sketch the principal angle with the terminal arm in the correct quadrant.
Draw a triangle by dropping a  to the x–axis.
Determine the related acute angle (& label the sides if it is a special triangle).
Use a calculator or the special triangle (for exact ratios) to determine the required
trigonometric ratio of the related acute angle.
5. Use the CAST Rule to determine whether the ratio is positive or negative.
Ex 
Evaluate. Leave answer as an exact value where appropriate.
a)
cos 330
b)
sin 315
c)
tan 150
d)
sin 225
e)
cot 120
f)
csc 240
Ex 
Use the related acute angle to state an equivalent expression:
cos 220o
Homework: p.299–300 #1, 2, 4, 5, 6abc and additional questions:
Evaluate. Leave answer as an exact value where appropriate.
Answers:
a) cos 155
b) sin 214
c) tan 315
d) cos 240
e) cot 150
f) cos 225
g) sin 300
h) sec 330
a) – 0.9063
b) – 0.5592
c) –1
d) − 2
e) −√3
f) −
√2
2
g) −
√3
2
1
h)
2√3
3