Download 7_5_Apply_the_Tangent_Ratio

5-MINUTE CHECK
SECTION 7.5 APPLY THE TANGENT RATIO



Students will analyze and apply the Tangent Ratio for
indirect measurement.
Why? So you can find the height of a roller coaster, as
seen in Ex 32.
Mastery is 80% or better on 5-minute checks and
practice problems.
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.
TRIGONOMETRIC RATIOS – SKILL DEVELOP

Let ∆ABC be a right
triangle. The since,
the cosine, and the
tangent of the acute
angle A are
defined as follows.
B
hypotenusec
A
b
side adjacent to angle A
Side
a opposite
angle A
C
Side adjacent to A b
cos A =
=
hypotenuse
c
sin A =
Side opposite A
hypotenuse
=
a
c
Side opposite A
a
tan A =
=
Side adjacent to A b
EX 1-FIND TANGENT RATIOS- SKILL DEVELOP
THINK……INK…..SHARE
EXAMPLE 2-FIND A LEG LENGTH-SKILL DEVELOP
EX 3….ESTIMATE HEIGHT USING TANGENT
REAL WORLD APPLICATION
EX…4 – SKILL DEVELOPMENT
EX 4 ….CONT
THINK….INK…SHARE
WHAT WAS THE OBJECTIVE FOR TODAY?
Students will analyze and apply the Tangent
Ratio for indirect measurement.
 Why? So you can find the height of a roller
coaster, as seen in Ex 32.
 Mastery is 80% or better on 5-minute checks
and practice problems.

HOMEWORK
Page 468-469
 # 3-29 odd

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
A
corresponding sides are
in proportion which
implies that the
trigonometric ratios for
A in each triangle are
the same.
B
17
8
C
15
B
8.5
4
A
7.5
C
EX. 1: FINDING TRIG RATIOS
Large
sin A =
cosA =
tanA =
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
Small
8 ≈ 0.4706
17
4 ≈ 0.4706
8.5
7.5 ≈ 0.8824
8.5
15 ≈ 0.8824
17
8 ≈ 0.5333
15
4 ≈ 0.5333
7.5
B
B
17
8.5
4
8
A
A
15
7.5
C Trig ratios are often
expressed as decimal
approximations.
C
EX. 2: FINDING TRIG RATIOS
S
sin S =
cos S =
tanS =
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
5 ≈ 0.3846
13
12 ≈ 0.9231
13
5 ≈ 0.4167
12
R
opposite
5
13 hypotenuse
T
12
adjacent
S
EX. 2: FINDING TRIG RATIOS—FIND THE SINE, THE
COSINE, AND THE TANGENT OF THE INDICATED
ANGLE.
R
sin S =
cosS =
tanS =
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
12 ≈ 0.9231
13
5 ≈ 0.3846
13
12 ≈ 2.4
5
R
adjacent
5
13 hypotenuse
T
12
opposite
S
EX. 3: FINDING TRIG RATIOS—FIND THE SINE,
THE COSINE, AND THE TANGENT OF 45
45
sin 45=
cos 45=
tan 45=
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
1
√2
1
√2
=
=
√2
2
√2
2
1
1
≈ 0.7071
≈ 0.7071
= 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 √2.
√2
hypotenuse
1
45
1
EX. 4: FINDING TRIG RATIOS—FIND THE SINE,
THE COSINE, AND THE TANGENT OF 30
30
sin 30=
cos 30=
tan 30=
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
1
= 0.5
2
√3
2
1
√3
≈ 0.8660
=
√3
3
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 √3 and the length
of the hypotenuse is 2.
≈ 0.5774
2
1
30
√3
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.
SAMPLE KEYSTROKES
Sample keystroke
sequences
74 sin
sin 74
Sample calculator display
Rounded
Approximation
0.961262695
0.9613
0.275637355
0.2756
3.487414444
3.4874
ENTER
74
COS
COS
74
ENTER
74
TAN
TAN
74
ENTER
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.
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
cos A
tan A =
B
c
A
a
b
C
USING TRIGONOMETRIC RATIOS IN REALLIFE

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.
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.
THE MATH
opposite
tan 59° =
Write the ratio
adjacent
h
tan 59° =
Substitute values
45
45 tan 59° = h
Multiply each side by 45
45 (1.6643) ≈ h
Use a calculator or table to find tan 59°
75.9 ≈ h
Simplify
The tree is about 76 feet tall.
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.
d
30°
76 ft
NOW THE MATH
opposite
sin 30° =
hypotenuse
76
sin 30° =
Write the ratio for
sine of 30°
d
30°
Substitute values.
d
d sin 30° = 76
76
d=
Divide each side by sin 30°
sin 30°
76
d=
Multiply each side by d.
0.5
d = 152
Substitute 0.5 for sin 30°
Simplify
A person travels 152 feet on the escalator stairs.
76 ft
REMINDERS:
After this section, you have a quiz on Thursday
or Friday.
 Chapter 9 exam will take place before you leave
for spring break . . . Take it before you go on
break.
 Binder check before spring break. These are
your new grades for the last quarter of the year.
Study and don’t slack off now.
