Transcript 8.5 The Tangent Ratio

Geometry
8.5
The Tangent Ratio
Trigonometry
The word trigonometry comes from the
Greek words that mean “triangle
measurement.”
In this course we will introduce the
trigonometry of right triangles.
opp.
adj.
Tangent ratio =
The tangent ratio is the ratio of the
length of the legs in a Rt. ∆
Tangent of <A:
opposite
leg
opposite leg
tan A 
adjacent leg
B
Tangent of <B:
opposite leg
tan B 
adjacent leg
adjacent leg
adjacent
leg
opposite leg
A
I. Express tan X and tan Y as ratios.
1.
2.
Z
9
Y
41
4.
2 3
40
21
X
3.
X
29
Y
4
Z
20
X
Z
Z
8
8
2
Y
X
Y
8 2
opp.
adj.
Exercises: tan =
1. Express tan X and tan Y as ratios.
Z
opp. 40
tan x 

adj.
9
opp.
9
tan y 

adj. 40
9
X
41
Y
opp.
adj.
Exercises: tan =
3. Express tan X and tan Y as ratios.
opp.
2  3
tan x 



adj. 2 3  3 
2 3
3
=
=
6
3
opp. 2 3
tan y 

 3
adj.
2
2√3
X
4
Z
2
Y
Answers to Exercises
2.
21
20
tan x =
tan y =
20
21
4.
tan x = 1 tan y = 1
Table of Trigonometric Ratios
• Turn to page 311 in your textbook.
• This table gives approximate decimal
values of the tangent ratio for some
angles.
• The table also can be used to find an
angle measure when given a tangent
value.
5. tan 15o = ______
6. tan 43o = ______
7. tan 82o = ______
8. tan ____ = .4663
9. tan _____ = 1.5399
10. tan _____ = 2.7475
Exercises
.2679
5. Complete: tan 15˚ ≈ _____
Step 1. Locate 15˚ in the angle column.
Step 2. Go across to the tangent column.
Step 3. Read the value: .2679
25˚ ≈ .4663
8. Complete: tan _____
Step 1. Locate .4663 in the tangent column.
Step 2. Go across to the angle column.
Step 3. Read the value: 25˚
Answers to Exercises
6. .9325
7. 7.1154
9.
57˚
10.
70˚
II. Find the value of x to the nearest tenth.
12.
11.
13.
17
25
x
14.
x
x
12
51
37
7.4
x
72
3
15.
16.
17
17. A freeway ramp has a 10% grade. What
angle does the ramp have with the ground?
[Note: Grade(%) = rise *100
y
x
8
200
x
run
60
z
Exercises
11. Find the value of x to the nearest tenth.
x
= tan 37°
25
x
= .7535
25
x  18.8
Exercises
13. Find the value of x to the nearest tenth.
7.4
tan 17° =
x
17
7.4
.3057 =
x
7.4
.3057x = 7.4
7.4
x=
.3057
x
x  24.2
Exercises
15. Find the value of x and y to the nearest tenth.
x
tan y° =
8
Solve for x :
Pythag. Thm.
x 2 + 82 = 17 2
x 2 + 64 = 289
x 2 = 225
x = 15
15
tan y° =
8
tan y° = 1.875
y  62°
Exercises
17. A freeway ramp has a 10% grade. What angle does
the ramp have with the ground?
1
ground
1
= .1
10
10
tan x°=.1
x6
Answers to Exercises
12. 9.2
14. 9.7
16. x ≈ 115.5, z ≈ 230.9
Homework
pg. 308 #1-24
Add hypotenuse, the three
proportions, Pythag and Converse,
multiply fractions vs. proportions,
45-45-90 and 30-60-90 formulas,
pythagorean triples to vocab list,
tangent