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Lesson 8-5
The Tangent Ratio
(page 305)
How can trigonometric ratios be used
to find sides and angles of a triangle?
Trigonometry, comes from 2 Greek words,
which mean “ triangle measurement .”
Our study of trigonometry will be limited to
Right Triangle Trigonometry.
The tangent ratio is the ratio
of the lengths of the legs .
B
c
A
a
b
leg
C
leg
In relationship to angle A …
c
A
B
a
b
adjacent leg
C
opposite leg
opposite leg vs adjacent leg
In relationship to angle B …
c
A
B
a
b
opposite leg
C
adjacent leg
opposite leg vs adjacent leg
Definition of Tangent Ratio
B
c
A
a
b
tangent of ∠A = tan A
C
a
BC
=
=
AC
b
length of the leg opposite ÐA
tan A =
length of the leg adjacent to ÐA
Definition of Tangent Ratio
B
c
A
a
b
tangent of ∠B = tan B
C
b
AC
=
=
BC
a
length of the leg opposite ÐB
tan B =
length of the leg adjacent to ÐB
B
c
a
A
C
b
length of the leg opposite ÐA
tan A =
length of the leg adjacent to ÐA
length of the leg opposite ÐB
tan B =
length of the leg adjacent to ÐB
opposite
tan =
Þ TOA
adjacent
remember
Example 1: Express tan A and tan B as ratios.
B
NOT
X
15
(a) tan A = ______
17
X
____
C
15
15
X
(b) tan B = ______
A
What can
we do now?
OH YEAH!
I know what I can do!
NEVER,
and I mean
NEVER, forget
my theorem!
Example 1: Express tan A and tan B as ratios.
8
15
(a) tan A = ______
B
reciprocals
8
____
17
15
8
(b) tan B = ______
BC + 15 = 17
A
2
15
BC + 225 = 289
2
BC = 64
Now we
BC = 64
can find the ratios!
Remember TOA.
BC = 8
2
C
2
2
Example 2
The table on page 311 gives approximate decimal
values of the tangent ratio for some angles.
“≈” means “is approximately equal to”
(a)
0.3640
tan 20º ≈ ____________
(b)
19.0811
tan 87º ≈ ____________
Now try this with a calculator!
To enter this in your calculator you will need
to use the TAN function key.
Enter TAN(20) then press ENTER (=)
and round to 4 decimal places.
(a)
0.3640
tan 20º ≈ ____________
Enter TAN(87) then press ENTER (=)
and round to 4 decimal places.
(b)
19.0811
tan 87º ≈ ____________
Example 3
The table on page 311 can also be used to find an
approximate angle measure given a tangent value.
“≈” means “is approximately equal to”
(a)
30º ≈ 0.5774
tan _______
(b)
tan _______
76º ≈ 4.0108
Now try this with a calculator!
To enter this in your calculator you will need to use
the inverse key or 2nd function key.
Enter TAN-1(.5774) then press ENTER (=)
and round to the nearest degree
(a)
30º ≈ 0.5774
tan _______
Enter TAN-1(4.0108) then press ENTER (=)
and round to the nearest degree
(b)
76º ≈ 4.0108
tan _______
Example 4 (a)
Find the value of x to the nearest tenth.
18.8
x ≈ ________
x
tan 37º =
25
25× tan37º= x
x
You can type this in your calculator!
37º
25
18.83 » x
Example 4 (b)
Find the value of x to the nearest tenth.
9.2
x ≈ ________
x
x
tan 72º =
3
3× tan72º= x
9.23 » x
72º
3
Example 4 (c)
Find the value of y to the nearest degree.
51º
y ≈ ________
5
tan yº =
4
æ5ö
y = tan ç ÷
è4ø
-1
5
yº
4
Type this in your calculator!
y » 51.3
Example 4 (d)
Find the value of y to the nearest degree.
32º
y ≈ ________
5
tan yº =
8
-1æ 5 ö
y = tan ç ÷
è 8ø
y » 32.0
x
8
yº
5
89
x +5 =
2
2
( 89 )
x 2 + 25 = 89
x 2 = 64
x =8
2
OPTIONAL Assignment
Written Exercises on pages 308 & 309
1 to 21 odd numbers
~ #22 is BONUS! ~
PK Hint:
For #19 on page 309 you should first read
Example 3 on page 306.
How can trigonometric ratios be used
to find sides and angles of a triangle?
The grade of a road is 8%.
What angle does the road make with the horizontal?
vertical
angle
horizontal
8
Grade =
100
The grade of a road is 8%.
What angle does the road make with the horizontal?
vertical
rise = 8
angle
horizontal
run = 100
8
Grade =
100
The grade of a road is 8%.
What angle does the road make with the horizontal?
vertical
rise = 8
angle
8
Grade =
100
horizontal
run = 100
8
tan "angle" =
100
-1 æ 8 ö
"angle" = tan ç
è 100 ÷ø
"angle" = 4.57
The road makes a 5º angle
with the horizontal.
Assignment:
TRIGONOMETRY WORKSHEET #1
This will be given after the next lesson.
Put #19 from page 309 on back of worksheet!
How can trigonometric ratios be used
to find sides and angles of a triangle?