MCR 3U U5
Trigonometric Ratios for Angles > 90
Once you know the trig ratios for angles between 0 and 90 you can use these to determine the trig values of
any angle > 90. The ratio values will either be the same or opposite to the values of those angles between 0
and 90. To these relationships we will first explore angles that are multiples of the frequently used angles 0,
45, 60, and 90.
The diagram shows the angle 45 drawn on a Cartesian plane. Draw the angles on the other Quadrants and
determine the Related Acute Angle. Use CAST or the fact that the x and y values are positive and/or negative in
the various quadrants (see below) to come up with the ratios of each angle.
opp y 1
 
hyp r
2
Cos 45 =
Sin 135 =
y
Sin 45 
y
Cos 135 =
2
Tan 45 =
2
1
x
45
1
Csc 45 =
Csc 135 =
Sec 45 =
Sec 135 =
Cot 45 =
Sin 225 =
Cot 135 =
Cos 225 =
x
45
1
Sin 315 =
y
1
Tan 135 =
y
Cos 315 =
2
Tan 225 =
2
1
x
45
1
Csc 225 =
Tan 315 =
Sec 315 =
Cot 225 =
Cot 315 =
x
45
1
Csc 315 =
Sec 225 =
1
To remember the sign of the trig ratio for angles in Quadrants 2, 3, and 4 we can use the CAST rule or
just use the trig ratios in this form: sin   y , cos   x , tan   y , csc  r , sec   r , cot   x
r
r
x
y
x
y
where (x,y) is the point on the terminal arm of the angle touching the circle and x2 + y2 = r2
The “CAST” Rule:




Quadrant 1
0     90 
y

sin  

r

x

cos  

r

y

tan  

x

All are +’ve
Quadrant 2
90     180 
y

sin  

r

x

cos  

r

y

tan  

x

Only
Sine is +’ve
Quadrant 3
180     270 
y

sin  

r

x

cos  

r

y

tan  

x

Only
Tosine is +’ve
Quadrant 4
270     360 
y

sin  

r

x

cos  

r

y

tan  

x

Only
Cosine is +’ve
Example 1: The point P(3, 6) lies on the terminal arm of an angle  in standard position. Draw a diagram to
represent this angle then determine the exact values of sin , cos  and tan .
Example 2: Find the exact values of the sine, cosine and tangent of 120 (this means do not use your
calculator).
Example 3: Evaluate each of the following (to 4 decimal places).
a) sin 238
d) csc 175
b) cos 312
e) sec 21
c) tan 197
f) cot 296
Example 4: Find the measure of the angle  to the nearest degree (0    360).
a) sin  = 
b) cos  = 0.7431
c) tan  = -14.3007
d) csc  = 
e) sec  = 1.0457
f) cot  = 1.1504
Practice:
1. The coordinates of a point P on a terminal arm of an angle  in standard position are given, where
0    360 . Draw a diagram to represent this angle, then determine the exact values of sin , cos  and tan
.
a) P(8, 15)
b) P(-3. 5)
c) P(-4, -3)
d) P(12, -5) e) P(-2, -7)
f) P(3, -2)
2. Find the exact value of each trigonometric ratio:
a) sin 45
b) cos 135
c) tan 225
e) cos 60
f) tan 120
g) sin 300
i) tan 30
j) sin 150
k) cos 210
d) sin 315
h) cos 240
l) tan 330
3. Evaluate each of the following (to 4 decimal places).
a) sin 172
d) csc 308
b) cos 211
e) sec 159
c) tan 284
f) cot 143
4. Find the measure of the angle  to the nearest degree (0    360).
a) sin  = 
b) cos  = 
c) tan  = 0.7813
d) csc  = 4.4454
e) sec  = 2.0627
f) cot  = 0.0699
Answers to Example Questions:
2
1
1. sin   
, cos   
, tan   2
5
5
3
1
2. sin 120  
, cos 120    , tan 120    3
2
2
3. a) 0.8480 b) 0.6691 c) 0.3057 d) 11.4737 e) 1.0711 f) 0.4877
4. a) 63 or 117 b) 42 or 318 c) 94 or 274 d) 203 or 337 e) 17 or 343 f) 41 or 221
Answers to Practice Questions:
15
5
15
8
5
3
1. a) sin  =
, cos =
, tan  =
b) sin  =
, cos = 
, tan  = 
8
3
17
17
34
34
3
3
5
4
5
12
c) sin  =  , cos =  , tan  =
d) sin  =  , cos =
, tan  = 
5
4
12
5
13
13
7
2
7
2
2
3
e) sin  = 
, cos = 
, tan  =
f) sin  = 
, cos =
, tan  = 
2
3
53
53
13
13
2. a)
1
b) 
1
c) 1 d) 
1
e)
3
3
1
1
1
1
1
f)  3 g) 
h) 
i)
j) k) 
l) 
2
2
2
2
2
3
3
2
2
2
3. a) 0.1392 b) 0.8572 c) 4.0108 d) 1.2690 e) 1.0711 f) 1.3270
4. a) 15 or 165 b) 14 or 346 c) 38 or 218 d) 193 or 347 e) 61 or 299 f) 94 or 274