MCT4C1
Day 5: Trigonometric Ratios of Any Angle Between 0 and 360
An angle in standard position has its initial arm on the positive x-axis. A positive angle is measured
counterclockwise.
The trigonometric ratios are expressed in terms of x, y, and r.
sin  
y
r
Example 1:
cos  
x
r
tan  
y
x
The point (5 , 12) is on the terminal arm of an angle  in standard position. Determine
the 3 primary trigonometric ratios for angle .
When the terminal arm of the angle is NOT in quadrant 1, construct a triangle by making a VERTICAL
line from the given point to the x-axis.
Example 2:
The point (-4 , 3) is on the terminal arm of an angle  in standard position. Determine the
three primary trigonometric ratios for angle .
MCT4C1
Example 3:
Draw and label the two special triangles
Example 4:
Redraw each of the triangles so that the hypotenuse is 1 unit long.
Example 5:
a) Use these new special triangles to determine the lengths of the sides of each triangle,
determine the coordinates of the end point on the terminal arm of each angle.
b) Determine the exact values for the sine, cosine and tangent ratio of each of the given
angles.
a)
b)

c)
 120
135
d)
e)
225


150
f)

210
240

MCT4C1
g)
h)
300
i)
330

315

Note: A unit circle is a circle whose radius is 1 unit.
Example 6:
Summarize your results on the given unit circle:

MCT4C1
HW – Day 5: Trigonometric Ratios of Any Angle Between 0 and 360
1. The coordinates of a point P on the terminal arm of each angle  in standard position are shown.
Determine the exact values of sin  , cos  , and tan  .
2. Each point lies on the terminal arm of angle  in standard position.
i) Draw a sketch of each angle  .
ii) Determine the exact value of r.
iii) Determine the 3 primary trigonometric ratios for angle  .
a) (5, 11)
b) (-8, 3)
c) (–5, -8)
d) (6, -8)
MCT4C1
3. For each trigonometric ratio, use a sketch to determine
i)
in which quadrant the terminal arm of the angle lies
ii)
the value of the related acute angle
iii)
the sign of the ratio
iv)
the exact value of the ratio
a) sin 315
b) tan 120
c) cos 240
d) tan 225
Answers:
15
8
15
, cos   , tan  
17
15
8
3
4
3
c) sin    , cos    , tan  
5
5
4
7
2
7
e) sin   
, cos   
, tan  
2
53
53
1. a) sin  
2.
a) ii) r  146 iii) sin  
11
, cos  
146
3
b) ii) r  73 iii) sin  
, cos  
73
8
c) ii) r  89 iii) sin  
, cos  
89
d) ii) r  10 iii) sin  
5
3
5
, cos   
, tan   
3
34
34
5
12
5
d) sin    , cos   , tan  
13
13
12
2
3
2
f) sin   
, cos  
, tan   
3
13
13
b) sin  
5
, tan  
146
8
3
, tan  
8
73
5
8
, tan  
5
89
4
3
4
, cos   , tan  
5
5
3
3. a) i) Q4 ii) 45° iii) - iv) sin 315  
1
2
b) i) Q2 ii) 60° iii) - iv) tan120   3
1
2
d) i) Q3 ii) 45° iii) + iv) tan 225  1
c) i) Q3 ii) 60° iii) - iv) cos 240 
11
5