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Chapter 5
Section 5.3
Evaluating Trigonometric Functions
Hypotenuse, Adjacent and Opposite sides of a Triangle
In a right triangle (a triangle with a right angle) the side that
does not make up the right angle is called the hypotenuse. For
an angle  that is not the right angle the other two sides are
names in relation to it. The opposite side is a side that makes
up the right angle that is across from . The adjacent side is the
side that makes up the right angle that also forms the angle .
hypotenuse
opposite side

adjacent side
hypotenuse

adjacent side
opposite side
The Trigonometric Ratios
hypotenuse
For any right triangle if we pick a certain angle  we can form six
opposite
side
different ratios of the lengths of the sides. They are the sine,

cosine, tangent, cotangent, secant and cosecant (abbreviated
adjacent side
sin, cos, tan, cot, sec, csc respectively).
opposite side
opposite side
hypotenuse
tan  
sin  
sec  
adjacent side
hypotenuse
adjacent side
hypotenuse
adjacent side
adjacent side
csc


cos  
cot  
opposite side
hypotenuse
opposite side
To find the trigonometric ratios when the lengths of the sides of a right triangle
are known is a matter of identifying which lengths represent the hypotenuse,
adjacent and opposite sides. In the triangle below the sides are of length 5, 12
and 13. We want to find the six trigonometric ratios for each of its angles  and .
sin 
5
13
cos 
12
13
tan 
5
12
cot 
12
5
sec 
13
12
csc 
13
5

sin 
12
13
cos 
5
13
12
tan 
12
5
Notice the following are equal:
cot 
5
12
sec 
13
5
csc 
13
12
13
5

sin   cos 
sin   cos 
tan   cot 
tan   cot 
sec   csc 
sec   csc 
The angles  and  are called complementary angles (i.e. they sum up to 90).
The “co” in cosine, cotangent and cosecant stands for complementary. They refer
to the fact that for complementary angles the complementary trigonometric
ratios will be equal. (i.e. sin 𝐴 = cos 90° − 𝐴 ,tan 𝐴 = cot 90° − 𝐴 , etc.)
Trigonometric Ratios of Special Angles
45-45-90 Triangles
30-60-90 Triangles
If you consider a square where
each side is of length 1 then the
diagonal is of length 2.
If you consider an equilateral triangle where each side is
of length 1 then the perpendicular to the other side is of
Ratios are:
2 2
:
:1
2 2
45
1
x 2
1
45
sin 45
cos 45
1
2

2
2
1
1
1
cot 45
(Ratios are
1 3
: : 1)
2 2
30
x
1
2

2
2
tan 45
length
3
.
2
1
1
1
sec 45
2

1
csc 45
2

1
1 1  x
2
11  x2
2  x2
2
2
60
1
2
2x
3
2
2
sin 60
3
2
cos 60
1
2
tan 60
cot 60
sec 60
csc 60
3
1
3

3
3
2
2
2 3

3
3
1
2
x 2   12   12
2
x 2  14  1
x2 
x
sin 30
1
2
cos 30
3
2
tan 30
1
3

3
3
cot 30
3
3
4
sec 30
3
2
csc 30
2
2 3

3
3
2
The values of the trigonometric functions will be the same as that of
the trigonometric ratios. In particular for the angles
0°,30°,45°,60°,90°. In the picture to the right the first quadrant is
shown along with the terminal points on the unit circle.
sin t cos t tan t cot t sec t csc t
0
0
1
0
30
1
2
3
2
3
3
45
2
2
2
2
1

3
2
1
2

1
0
60
90
3
-
-
1
-
3
2 3
3
2
1
2
2
3
3
2
2 3
3
0
-
1
t  90 , P(0,1)
 
t  60 , P 12 ,
3
2
t  45 , P

2
2
,
t  30, P
2
2

 ,
3
2
1
2
t  0 , P(1,0)
Reference Angles
The reference angle for an angle is the angle made when you drop a line straight
down to the x-axis. it is the angle made by the x-axis regardless of what side of it
you are on. (Go to closest multiple of 180° and add or subtract.)
225
120
330
60
60
30
45
-300
240
Helps to find the values.
Find the six trigonometric − 3
2
ratios for 240° angle.
60
−1
2
− 3
=
2
tan 240° = 3
sin 240°
sec 240° = −2
−1
=
2
1
°
cot 240 =
3
−2
°
csc 240 =
3
cos 240°
Find the values for a, b, c, and d in
the triangles pictures to the right.
45°
24
c
To find a:
b
60°
a
𝑎
0
cos 60 =
24
1
𝑎
=
2 24
𝑎 = 12
d
To find c:
𝑏
=
𝑐
2 12 3
=
2
𝑐
cos 450
To find b:
𝑏
=
24
3
𝑏
=
2
24
𝑏 = 12 3
sin 60°
To find d:
𝑐=
24 3
2
tan 45°
1=
𝑑
=
𝑏
𝑑
12 3
𝑑 = 12 3