9.5 Trigonometric Ratios

1
9.5 Trigonometric Ratios

• Geometry

2
Objectives/Assignment

• Find the since, the cosine, and the tangent of an
  acute triangle.
• Use trionometric ratios to solve real-life
  problems, such as estimating the height of a tree
  or flagpole.
• To solve real-life problems such as in finding
  the height of a water slide.

3
Finding Trig 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.

4
Trigonometric Ratios

• Let ?ABC be a right triangle. The since, the
  cosine, and the tangent of the acute angle ?A are
  defined as follows.

Side adjacent to ?A
b
cos A

hypotenuse
c
Side opposite ?A
a
sin A

hypotenuse
c
Side opposite ?A
a
tan A

Side adjacent to ?A
b
5
Note

• The value of a trigonometric ratio depends only
  on the measure of the acute angle, not on the
  particular right triangle that is used to compute
  the value.

6
Ex. 1 Finding Trig Ratios

• Compare the sine, the cosine, and the tangent
  ratios for ?A in each triangle beside.
• By the SSS Similarity Theorem, the triangles are
  similar. Their corresponding sides are in
  proportion which implies that the trigonometric
  ratios for ?A in each triangle are the same.

7
Ex. 1 Finding Trig Ratios
Large Small



opposite
8
sin A

0.4706
4

0.4706
hypotenuse
17
8.5
adjacent
7.5
cosA
15

0.8824

0.8824
hypotenuse
8.5
17
opposite
tanA
8
4

0.5333

0.5333
adjacent
15
7.5
Trig ratios are often expressed as decimal
approximations.
8
Ex. 2 Finding Trig Ratios
?S



opposite
5
sin S

0.3846
hypotenuse
13
adjacent
cosS
12

0.9231
hypotenuse
13
opposite
tanS
5

0.4167
adjacent
12
9
Ex. 2 Finding Trig RatiosFind the sine, the
cosine, and the tangent of the indicated angle.
?R



opposite
12
sin S

0.9231
hypotenuse
13
adjacent
cosS
5

0.3846
hypotenuse
13
opposite
tanS
12

2.4
adjacent
5
10
Ex. 3 Finding Trig RatiosFind the sine, the
cosine, and the tangent of 45?
45?



opposite
1
v2
sin 45?


0.7071
hypotenuse
v2
2
adjacent
1
v2
cos 45?


0.7071
hypotenuse
v2
2
opposite
1
tan 45?
adjacent

1
1
Begin by sketching a 45?-45?-90? triangle.
Because all such triangles are similar, you can
make calculations simple by choosing 1 as the
length of each leg. From Theorem 9.8 on page
551, it follows that the length of the hypotenuse
is v2.
v2
45?
11
Ex. 4 Finding Trig RatiosFind the sine, the
cosine, and the tangent of 30?
30?



opposite
1
sin 30?

0.5
hypotenuse
2
adjacent
v3
cos 30?

0.8660
hypotenuse
2
opposite
v3
1
tan 30?

adjacent

0.5774
3
v3
Begin by sketching a 30?-60?-90? triangle. To
make the calculations simple, you can choose 1 as
the length of the shorter leg. From Theorem 9.9,
on page 551, it follows that the length of the
longer leg is v3 and the length of the hypotenuse
is 2.
30?
v3
12
Ex 5 Using a Calculator

• You can use a calculator to approximate the sine,
  cosine, and the tangent of 74?. Make sure that
  your calculator is in degree mode. The table
  shows some sample keystroke sequences accepted by
  most calculators.

13
Sample keystrokes
Sample keystroke sequences Sample calculator display Rounded Approximation
74 74 0.961262695 0.9613
0.275637355 0.2756
3.487414444 3.4874
sin
sin
ENTER
74 74
COS
COS
ENTER
74 74
TAN
TAN
ENTER
14
Notes

• If you look back at Examples 1-5, you will notice
  that the sine or the cosine of an acute triangles
  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 or a right triangle is always
  less than the length of its hypotenuse, so the
  ratio of these lengths is always less than one.

15
Trigonometric Identities

• A trigonometric identity is an equation involving
  trigonometric ratios that is true for all acute
  triangles. You are asked to prove the following
  identities in Exercises 47 and 52.

• (sin A)2 (cos A)2 1

sin A
tan A
cos A
16
Using Trigonometric Ratios in Real-life

• Suppose you stand and look up at a point in the
  distance. Maybe you are looking up at the top of
  a tree as in Example 6. The angle that your line
  of sight makes with a line drawn horizontally is
  called angle of elevation.

17
Ex. 6 Indirect Measurement

• You are measuring the height of a Sitka spruce
  tree in Alaska. You stand 45 feet from the base
  of the tree. You measure the angle of elevation
  from a point on the ground to the top of 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.

18
The math
Write the ratio
Substitute values
Multiply each side by 45
45 tan 59 h
Use a calculator or table to find tan 59
45 (1.6643) h
Simplify
75.9 h
?The tree is about 76 feet tall.
19
Ex. 7 Estimating 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 of 76
  feet.

30
20
Now the math
30
Write the ratio for sine of 30
Substitute values.
d sin 30 76
Multiply each side by d.
Divide each side by sin 30
Substitute 0.5 for sin 30
d 152
Simplify
?A person travels 152 feet on the escalator
stairs.