Download 9.5 Trigonometric Ratios

Finding Trigonometric Ratios
A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.
The word trigonometry is derived from the ancient Greek language and means
measurement of triangles.
The three basic trigonometric ratios are sine, cosine, and tangent, which are
abbreviated as sin, cos, and tan, respectively.
Finding Trigonometric Ratios
A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.
TRIGONOMETRIC RATIOS
Let ABC be a right triangle. The sine, the cosine, and the tangent of the
acute angle A are defined as follows.
a
side opposite A
=
sin A =
c
hypotenuse
cos A =
side adjacent A
b
=
hypotenuse
c
a
side opposite A
tan A =
=
b
side adjacent to A
B
hypotenuse
c
A
side
a opposite
A
C
b
side adjacent to A
The value of the trigonometric ratio depends only on the measure of the acute
angle, not on the particular right triangle that is used to compute the value.
Finding Trigonometric Ratios
Compare the sine, the cosine, and the tangent ratios for A in each triangle below.
B
SOLUTION
8.5
By the SSS Similarity Theorem, the triangles are similar.
4
C
A
7.5
B
Their corresponding sides are in proportion, which implies that
the trigonometric ratios for  A in each triangle are the same.
17
8
A
15
sin A =
cos A =
tan A =
opposite
hypotenuse
Large triangle
8
 0.4706
17
adjacent
hypotenuse
opposite
adjacent
15
 0.8824
17
8
 0.5333
15
Small triangle
4
 0.4706
8.5
7.5
 0.8824
8.5
4
 0.5333
7.5
C
Trigonometric ratios
are frequently
expressed as decimal
approximations.
Finding Trigonometric Ratios
Find the sine, the cosine, and the tangent of the indicated angle.
R
S
13
5
SOLUTION
T
S
12
The length of the hypotenuse is 13. For S, the length of the opposite side
is 5, and the length of the adjacent side is 12.
sin S =
cos S =
opp.
5
=
hyp.
13
 0.3846
adj.
12
 0.9231
=
hyp.
13
opp.
5
 0.4167
tan S =
=
adj.
12
R
13
5
opp.
T
hyp.
12 adj.
S
Finding Trigonometric Ratios
Find the sine, the cosine, and the tangent of the indicated angle.
R
R
13
5
SOLUTION
T
S
12
The length of the hypotenuse is 13. For R, the length of the opposite side
is 12, and the length of the adjacent side is 5.
sin R =
cos R =
opp. 12
=
hyp.
13
 0.9231
adj.
5
=
 0.3846
hyp.
13
opp.
12
tan R =
=
= 2.4
adj.
5
R
13
5
adj.
T
hyp.
12 opp.
S
Trigonometric Ratios for 45º
Find the sine, the cosine, and the tangent of 45º.
SOLUTION
Because all such triangles are similar, you can make calculations simple by
choosing 1 as the length of each leg.
From the 45º-45º-90º Triangle Theorem, it follows that the length of the
hypotenuse is 2 .
sin 45º =
opp. =
hyp.
1
=
2
2
2
 0.7071
2
1
cos 45º =
adj. =
hyp.
1
=
2
2
2
 0.7071
45º
1
tan 45º =
opp.
1
=
adj.
1
=1
hyp.
Trigonometric Ratios for 30º
Find the sine, the cosine, and the tangent of 30º.
SOLUTION
To make the calculations simple, you can choose 1 as the length of the shorter leg.
From the 30º-60º-90º Triangle Theorem, it follows that the length of the longer leg is
3 and the length of the hypotenuse is 2.
sin 30º =
cos 30º =
tan 30º =
1
opp.
=
= 0.5
2
hyp.
adj.
=
hyp.
opp. =
adj.
2
3
2
1
3
 0.8660
=
3
3
1
30º
 0.5774
3
Using a Calculator
You can use a calculator to approximate the sine, the cosine, and the tangent
of 74º. Make sure your calculator is in degree mode. The table shows some
sample keystroke sequences accepted by most calculators.
Sample keystroke sequences
Sample calculator
display
Rounded
approximation
74
sin
or
sin
74
Enter
0.961261695
0.9613
74
cos
or
cos
74
Enter
0.275637355
0.2756
74
tan
or
tan
74
Enter
3.487414444
3.4874
Finding Trigonometric Ratios
The sine or cosine of an acute angle is always less than 1.
The reason is that these trigonometric ratios involve the ratio of a leg of a right
triangle to the hypotenuse.
The length of a leg of a right triangle is always less than the length of its
hypotenuse, so the ratio of these lengths is always less than one.
Because the tangent of an acute angle involves the ratio of one leg to another leg,
the tangent of an angle can be less than 1, equal to 1, or greater than 1.
Finding Trigonometric Ratios
The sine or cosine of an acute angle is always less than 1.
The reason is that these trigonometric ratios involve the ratio of a leg of a right
triangle to the hypotenuse.
The length of a leg of a right triangle is always less than the length of its
hypotenuse, so the ratio of these lengths is always less than one.
Because the tangent of an acute angle involves the ratio of one leg to another leg,
the tangent of an angle can be less than 1, equal to 1, or greater than 1.
TRIGONOMETRIC IDENTITIES
A trigonometric identity is an equation involving trigonometric ratios that is
true for all acute angles. The following are two examples of identities:
B
(sin A) 2 + (cos A) 2 = 1
tan A =
sin A
cos A
c
a
C
A
b
Using Trigonometric Ratios in Real Life
Suppose you stand and look up at a point in the distance, such as the top of
the tree. The angle that your line of sight makes with a line drawn horizontally
is called the angle of elevation.
Indirect Measurement
FORESTRY You are measuring the height of a Sitka spruce tree in Alaska.
You stand 45 feet from the base of a tree. You measure the angle of elevation
from a point on the ground to the top of the tree to be 59°. To estimate the
height of the tree, you can write a trigonometric ratio that involves the height h
and the known length of 45 feet.
tan 59° =
opposite
adjacent
Write ratio.
tan 59° =
opposite
h
adjacent
45
Substitute.
45 tan 59° = h
Multiply each side by 45.
45(1.6643)  h
Use a calculator or table to find tan 59°.
74.9  h
Simplify.
The tree is about 75 feet tall.
Estimating a Distance
ESCALATORS The escalator at the Wilshire/Vermont Metro Rail Station in
Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels
on the escalator stairs, you can write a trigonometric ratio that involves the
hypotenuse and the known leg length of 76 feet.
sin 30° =
opposite
hypotenuse
Write ratio for sine of 30°.
sin 30° =
76opposite
hypotenuse
d
Substitute.
d sin 30° = 76
76
d=
sin 30°
76
d=
0.5
d = 152
Multiply each side by d.
d
76 ft
Divide each side by sin 30°.
30°
Substitute 0.5 for sin 30°.
Simplify.
A person travels 152 feet on the escalator stairs.