Transcript No Slide Title - Coweta County Schools

• How do I use the
sine, cosine, and
tangent ratios to
solve triangles?
5.2
Apply the Tangent Ratio
Trigonometric Ratios
B
c
a
C
b
opposite
a
sin A 
=
hypotenuse
c
A
b
adjacent
=
cos A =
hypotenuse c
opposite a
tan A =
=
adjacent b
5.2
Apply the Tangent Ratio
Tangent Ratio
Let rABC be a right triangle with acute A.
The tangent of A (written as tan A) is
defined as follows:
B
C
A
BC
length of leg opposite A
tan A 

AC
length of leg adjacent t o A _______
5.2
Apply the Tangent Ratio
Example 1 Find tangent ratios
Find tan X and tan Y. Write each
answer as a fraction and as a
decimal rounded to four places.
Y
8
Z
17
15
0.5333
 ______
X
YZ
8
opp. X
tan X 


XZ _______
15
adj. to X _______
XZ
15
opp. Y
1.875
tan Y 


 ______
YZ _______
8
adj. to Y _______
In the right triangle, rXYZ, X and Y are
_______________
complementary angles. You can see that the
tangent ratios of the _________________
angles are
complementary
___________.
reciprocals
5.2
Apply the Tangent Ratio
Example 2 Find a leg length
Find the value of x.
Use the tangent of an acute angle
to find a leg length.
opp.
o
tan 25 
__________
adj.
x
o
tan 25 
_______
12
o
___

tan
25
x
12
___
0.4663  x
12  ______
_____
5.6  x
x
25o
12
Write a ratio for tangent of 25o.
Substitute.
Multiply each side by ____.
12
Use a calculator to find tan 25o.
Simplify.
5.2
Apply the Tangent Ratio
Checkpoint. Complete the following exercises.
1. Find tan A and tan B. Round your
answer to four decimal places.
A
tan A  18
24
 0.75
tan B  24  1.3333
18
C
24
18
30
B
5.2
Apply the Tangent Ratio
Checkpoint. Complete the following exercises.
2. Find the value of x. Round to the
nearest tenth.
x
38o
20
x
tan 38 
20
o
20
20
x  20  0.7813
x 15.6
5.2
Apply the Tangent Ratio
Example 3 Compare the tangent ratios for similar triangles
Find tan X and tan Y for similar triangles. Then compare the
Y
tangent ratios.
Y
b
Z
c
a
3b
X
b
tan X 
a
_______
a
tan Y 
b
______
3c
X
3a
tan X  3b  b
a
_______
______
3a
tan Y  3a  a
______
3b ______
b
Z
equivalent
The values of tan X and tan Y for the similar triangles are ________.
5.2
Apply the Tangent Ratio
Example 4 Estimate height using tangent
Find the height h of the flagpole to the
nearest foot.
opp.
tan 70 
adj.
_______
o
tan 70o  h
7
_______
___
7  tan 70o  ___
h
___
19  h
h
Write ratio for
tangent of 70o.
Substitute.
70 o
7 ft
Multiply each side by ___.
7
Use a calculator to simplify.
19 feet tall.
The flagpole is about _____
5.2
Apply the Tangent Ratio
Checkpoint. Complete the following exercises.
3. Find the height h of the flagpole
in Example 4 to the nearest foot if
the angle is 75o.
7
h
tan 75 
7
o
7
h  7  3.7321
h  26 ft
h
75o
7 ft
5.2
Apply the Tangent Ratio
Pg. 173, 5.2 #1-25
5.3
Apply the Sine and Cosine Ratios
Pg. 218, 6.4 #1-32