Transcript Sec 3.2

Chapter 3
Trigonometric Functions of Angles
Section 3.2
Trigonometry of Right Triangles
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
For any right triangle if we pick a certain angle  we
can form six different ratios of the lengths of the sides.
They are the sine, cosine, tangent, cotangent,
secant and cosecant (abbreviated sin, cos, tan, cot,
sec, csc respectively).
hypotenuse
opposite
side

adjacent side
sin  
opposite side
hypotenuse
tan  
opposite side
adjacent side
sec  
hypotenuse
adjacent side
cos  
adjacent side
hypotenuse
cot  
adjacent side
opposite side
csc  
hypotenuse
opposite side
To find the trigonometric ratios when the lengths of the sides of a right triangle are
know 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

13
5

12
Notice the following are equal:
sin   cos 
sin   cos 
tan   cot 
tan   cot 
sec   csc 
sec   csc 
sin 
12
13
cos 
5
13
tan 
12
5
cot 
5
12
sec 
13
5
csc 
13
12
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.
Pythagorean Theorem and Trigonometric Ratios
c
The Pythagorean Theorem relates the sides of a right
triangle so that if you know any two sides of the triangle you
can find the remaining one. This is particularly useful in trig
since two sides will then determine all six trigonometric
ratios.
b
a
a2  b2  c2
Determine the six trigonometric ratios for the right triangle pictured below.
sin 
cos 
tan 
13
7
6
7
cot 
13
6
6
13
sec 
7
6
csc 
7
13

x
7

sin 
6
7
cos 
13
7
6
13
tan 
6
cot 
First we need to
determine the length of
the remaining side
which we will call x and
apply the Pythagorean
Theorem.
x 2  62  7 2
x 2  36  49
x 2  13
x  13
sec 
13
6
7
13
csc 
7
6
Finding Other Trigonometric Ratios by Knowing One
If one of the trigonometric ratios is known it is possible to find the other five
trigonometric ratios by constructing a right triangle with an angle and sides
corresponding to the ration given. For example, if we know that the sin  = ¾ find
the other trigonometric ratios.
1. Make a right triangle and
label one angle .
2. Make the hypotenuse
length 4 and the opposite
side length 3.
3. Find the length of the
remaining side.
4. Find the other
trigonometric ratios.
4
3

x= 7
sin 
3
4
cos 
7
4
3
7
tan 
x 2  32  4 2
x 2  9  16
x2  7
x 7
cot 
sec 
7
3
4
7
csc 
4
3
Similar Triangles and Trigonometric Ratios
Triangles that have the exact same angles measures but whose sides can be of
different length are called similar triangles. Similar triangles have sides that are
proportional. That is to say the sides are just a multiple of each other. Consider the
example above where the sides of one triangle are just three times longer than the
side of the other triangle.
5
3

15
4
9
sin 
9
3

15 5
cos 
12
4

15
5
tan 
9
3

12
4
cot 
12
4

9
3
sec 
15
5

12
4
csc 
15 5

9
3

12
Similar Triangles have
equal trigonometric ratios!
Many of the ideas in trigonometry are based
on this concept.
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 length 3 .
1 1  x
2
45
1
x 2
2
11  x2
2
x 2   12   12
2
2
1
2  x2
45
1
sin 45
tan 45
1
2

2
2
1
1
1
cot 45
1
1
1
sec 45
2

1
csc 45
2

1
cos 45
x
x2 
3
2
x
60
1
2
2x
1
2

2
2
x 2  14  1
30
3
4
3
2
sin 60
3
2
sin 30
1
2
cos 60
1
2
cos 30
3
2
tan 60
3
tan 30
1
3

3
3
3
cot 60
1
3

3
3
cot 30
2
sec 60
2
sec 30
2
csc 60
2
2 3

3
3
csc 30
2
2 3

3
3
2
Finding the Length of Sides of Right Triangles
If you know the length of one of the sides and the measure of one of the angles of a
right triangle you can find the length of the other sides by using trigonometric ratios.
8
x
z
17
37
y
68
x
Find the values for x and y.
Find the values for x and z.
x
 sin 37 
8
x  8  sin 37 
x = 4.81452
x
 cot 68
17
x  17  cot 68
y
 cos 37
8
y = 6.38908

y  8  cos 37
z
 csc 68
17
z  17  csc 68
x = 6.86845
z = 18.3351