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
11 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
2x
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