Transcript Lesson 3 - Coweta County Schools

• How do I use the
sine, cosine, and
tangent ratios to
solve triangles?
5.3
Apply the Sine and Cosine Ratios
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
5.3
Apply the Sine and Cosine Ratios
Sine and Cosine Ratios
Let rABC be a right triangle with acute A.
The sine of A and cosine of A(written sin A
and cos A) is defined as follows:
B
C
A
BC
length of leg opposite A
sin A 

AB
length of hypotenuse
_______
length of leg adjacent t o A AC
cos A 

AB
length of leg hypotenuse
_______
5.3
Apply the Sine and Cosine Ratios
Example 1 Find sine ratios
Find sin X and sin Y. Write each
answer as a fraction and as a
decimal rounded to four places.
Y
7
Z
25
24
7
opp. X YZ
0.28
sin X 


 ______
XY _______
25
hyp
_______
24
opp. Y XZ
sin Y 


 ______
0.96
XY _______
25
hyp
_______
X
5.3
Apply the Sine and Cosine Ratios
Example 2 Find cosine ratios
Find cos X and cos Y. Write each
answer as a fraction and as a
decimal rounded to four places.
Y
7
Z
25
24
24
adj. to X XZ
0.96
cos X 


 ______
XY _______
25
hyp
_______
7
adj. to Y YZ
0.28
cos Y 


 ______
XY _______
25
hyp
_______
X
5.3
Apply the Sine and Cosine Ratios
Checkpoint. Find the indicated measure.
Round to 4 decimal places, if necessary.
1. Find sin A and sin B.
sin A  21  0.7241
29
sin B  20  0.6897
29
2. Find cos A and cos B.
20
cos A 
 0.6897
29
21
cos B 
 0.7241
29
B
21
C
29
20
A
5.3
Apply the Sine and Cosine Ratios
Example 3 Use trigonometric ratios to find side lengths
Use a trigonometric ratio to find the
value of x in the diagram. Round to
the nearest tenth.
adj.
a. cos 31 
__________
hyp.
12
o
cos 31 
_______
x
1
12
x
o
cos 31
__________
x  12
__________
0.8572
a.
x
31o
o
12
x cos 31  12
o
o
cos 31 cos 31
o
14.0
x  _____
5.3
Apply the Sine and Cosine Ratios
Example 3 Use trigonometric ratios to find side lengths
Use a trigonometric ratio to find the
value of x in the diagram. Round to
the nearest tenth.
opp.
b.
sin 44 
hyp.
__________
x
o
sin 44 
_______
48
48  sin 44o  x
___
___
______
48 0
.6947   x
_____
33.3  x
o
b.
48
44 o
x
5.3
Apply the Sine and Cosine Ratios
Example 4 Sine and cosine ratios for similar triangles
Find the sine and cosine of
X , Y , L, and M of the
similar triangles. Then
compare the ratios.
M
Y
b
Z
c
a
3b
3c
X
b
a
N
sin X 
cos X 
c
c
_______
_______
sin Y  a
cos Y  b
c
c
______
______
sin L  3b  b
cos L  3a  a
c
c
3c
3c
______
______
______
______
a
b
3a
3b
sin M 

cos M 

c
c
3c
3c
______
______
______
______
3a
L
5.3
Apply the Sine and Cosine Ratios
Example 4 Sine and cosine ratios for similar triangles
Find the sine and cosine of
X , Y , L, and M of the
similar triangles. Then
compare the ratios.
M
Y
b
Z
c
a
3b
3c
X
L
3a
complementary
In rXYZ, X and Y are _______________
Y and sin Y = cos ___.
X
angles, so sin X = cos ___
complementary
In rLMN, L andM are _______________
M and sin M = cos ___.
L
angles, so sin L = cos ___
similar triangles,
Because rXYZ and rLMN are _______
L cos X = cos ___,
M and
L sin Y = sin ___,
sin X = sin ___,
cos Y = cos ___.
M
N
5.3
Apply the Sine and Cosine Ratios
Example 5 Use trigonometric ratios to find side lengths
Find the height of the parking ramp shown.
opp.
sin 27 
hyp.
_______
sin 27 o  x
65
_______
o
65  sin 27  x
___
___
______
65 0
.4540   x
_____
29.5  x
o
65 ft
27 o
x ft
5.3
Apply the Sine and Cosine Ratios
Checkpoint. Complete the following exercises.
3. Find the value of x. Round to the
nearest tenth.

x
28 sin 46  28
o
28
x
x  28  0.7193
x  20.1
28
y
46 o
5.3
Apply the Sine and Cosine Ratios
Checkpoint. Complete the following exercises.
4. Find the value of y. Round to the
nearest tenth.

y
28 cos 46  28
o
28
x
y  28 0.6947
y  19.5
28
y
46 o
5.3
Apply the Sine and Cosine Ratios
Pg. 180, 5.3 #1-19