TF.02.2 - Trigonometric
Ratios of Special Angles
MCR3U - Santowski
1
(A) 45-45-90 Triangles
• If we have an isosceles triangle and
make each non-hypotenuse side 1 unit long,
then we have the hypotenuse as 2
•
Each of our “base” angles are then 45 degrees
45
•
•

2
Then sin(45) = 1/ 2 = 2 / 2
Then cos(45)= 1/ 2 = 2 / 2
1
90
•
Then tan(45)= 1/1 = 1
45
1
2
(B) 45-45-90 Triangles on the Cartesian
Plane
•
we can now put our 45-45-90 triangle into the Cartesian plane and
investigate the trigonometric ratios of other key angles like 135,
225, 315, etc..
1
0.9
y
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
1
1.2
1.4
y
-0.1
-0.2
0.3
0.2
0.1
-1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
1
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
3
(C) 30-60-90 Triangles
•
To work with the 30-60-90 triangle, we will start with an
equilateral triangle, where each angle is 60 and we will set each
side to be 2 units long.
•
From one vertex, we will simply drop an altitude (to create a right
angle) to the opposite side, thereby bisecting the opposite side
into two equal lengths of 1 unit
•
The altitude has also bisected the top angle (so we now have a 30
degree angle)
•
So the altitude that we just drew measures (22 – 12) or 3 unit
•
So now the ratios can be determined
4
(D) Ratios in the 30-60-90 Triangle
•
sin 30 = ½
•
•
cos 30 = 3/2
tan 30 = 1/ 3 = 3/3
•
•
•
sin 60 = 3/2
cos 60 = ½
tan 60 = 3/1 = 3
30 30
2
2

3
90
60
1
60
1
5
(E) 30-60-90 Triangles on the Cartesian
Plane
•
we can now put our 30-60-90 triangle into the Cartesian plane and
investigate the trigonometric ratios of other key angles like 120,
150, 210, 240, 300, 330 etc..
y
0.9
0.3
y
0.8
0.2
0.7
0.1
-1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
0.6
1
-0.1
0.5
-0.2
0.4
y
-0.3
0.3
-0.4
0.2
-0.5
0.6
0.1
-0.6
0.5
-0.7
-0.6
-0.8
-0.4
-0.2
0.2
0.4
0.6
0.8
1
1.2
1.4
-0.1
0.4
-0.2
0.3
-0.3
0.2
0.1
-0.4
-0.2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
6
(F) Trig Ratios of Angles of Multiples of
90
To understand the trig ratios of angles
of 90, 180, 270, 360, etc, we will
simply go back to the Cartesian plane
and work with angles from ordered pairs
on the Cartesian plane
•
•
•
•
90 degrees  let’s put a point at (0,3)
180 degrees let’s put a point at (-2,0)
270 degrees  point at (0,-4)
360 degrees (or 0 degrees)  point at (1,0)
•
And recall that sin(A) = y/r, cos(A) = x/r and tan(A) = y/x
7
(F) Trig Ratios of Angles of Multiples of
90
•
•
•
sin 0 = 0/r = 0
cos 0 = r/r = 1
tan 0 = 0/r = 0
•
•
•
sin 90 = 1
cos 90 = 0
tan 90 = undefined
•
•
•
sin 180 = 0
cos 180 = -1
tan 180 = 0
•
•
•
sin 270 = -1
cos 270 = 0
tan 270 = undefined
•
•
•
sin 360 = sin 0 = 0
cos 360 = cos 0 = 1
tan 360 = tan 0 = 0
8
(G) Examples
• Ex 1. Determine the sine of 495°
• (i) first we determine what quadrant the angle
lies in  495°-360° = 135° which is thus a second
quadrant angle
• (ii) then subtract 180° - 135° = 45°, so we have a
45° angle in the second quadrant
• (iii) now, simply recall sin(45°) = 1/2
• (iv) now account for the quadrant, as the sine
ratio is positive in the second quadrant  so the
final answer is +1/ 2
9
(G) Examples
• Evaluate cos(-150°)
• (i) first we determine what quadrant the angle
lies in  360°+(-150°) = 210° which is thus a third
quadrant angle
• (ii) then subtract 210° - 180° = 30°, so we have a
30° angle in the second quadrant
• (iii) now, simply recall cos(30°) = 3/2
• (iv) now account for the quadrant, as the cosine
ratio is negative in the third quadrant  so the
final answer is - 3/2
10
(H) Internet Links
• Now try a couple of on-line “quizzes” to
see how well you understand the trig.
ratios of standard special angles:
• Introductory Exercises from U. of Sask
EMR
• Moderate Exercises from U. of Sask EMR
• Advanced Exercises from U. of Sask EMR
11
(I) Classwork & Homework
• Nelson text, Page 532, Q1-4,6,812eol,
12