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• How do I use the sine, cosine, and tangent ratios to solve triangles?
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Trigonometric Ratios
=a
c
A
B
C
a
b
c
=b
c
=a
b
cos A =adjacent
hypotenuse
tan A =opposite
adjacent
hypotenuse
oppositeAsin
5.3 Apply the Sine and Cosine Ratios
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5.3 Apply the Sine and Cosine Ratios
Sine and Cosine Ratios
A
B
C
_______hypotenuse oflength
A opposite leg oflength Asin
AB
BC
Let ABC be a right triangle with acute The sine of and cosine of (written sin A and cos A) is defined as follows:
AA.
A
_______hypotenuse leg oflength
A oadjacent t leg oflength A cos
AB
AC
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5.3 Apply the Sine and Cosine Ratios
Example 1 Find sine ratios Find sine ratios
Find sin X and sin Y. Write each answer as a fraction and as a decimal rounded to four places.
X
Y
Z 24
725
______hyp
X opp.Xsin
______________
XY
YZ
25
728.0
______hyp
Y opp.Y sin
______________
XY
XZ
25
2496.0
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5.3 Apply the Sine and Cosine Ratios
Example 2 Find cosine ratios Find cosine ratios
Find cos X and cos Y. Write each answer as a fraction and as a decimal rounded to four places.
X
Y
Z 24
725
______hyp
X toadj.X cos
______________
XY
XZ
25
2496.0
______hyp
Y toadj.Y cos
______________
XY
YZ
25
728.0
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5.3 Apply the Sine and Cosine Ratios
A
B
C 20
2129
Checkpoint. Find the indicated measure. Checkpoint. Find the indicated measure. Round to 4 decimal places, if necessary.Round to 4 decimal places, if necessary.1. Find sin A and sin B.
A sin 21
297241.0
B sin 20
296897.0
2. Find cos A and cos B.
A cos 2029
6897.0
B cos 21
297241.0
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5.3 Apply the Sine and Cosine Ratios
Example 3 Use trigonometric ratios to find side lengths 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.
12
x
o31
a.
__________
o31 cos a. hyp.
adj.
_______
o31 cos x
12
1
o31cosx 12o31cos o31cos
__________
xo31cos
12
__________
x8572.0
12 _____x 0.14
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5.3 Apply the Sine and Cosine Ratios
Example 3 Use trigonometric ratios to find side lengths 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.
__________
o44sin b. hyp.
opp.
_______
o44sin 48
x
x o44sin___48 x______ ___48
x_____3.33
48 xo44
b.
6947.0
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5.3 Apply the Sine and Cosine Ratios
Find the sine and cosine of , , , LYX and M of the
similar triangles. Then compare the ratios.
L
M
N a3
b3c3
X
Y
Z a
bc
_______
Xsin c
b
______
Ysin c
a
____________
Lsin c
b
3
3
____________
Msin c
a
3
3c
b
c
a
_______
X cos c
a
______
Y cos c
b
____________
L cos c
a
3
3
____________
M cos c
b
3
3c
a
c
b
Example 4 Sine and cosine ratios for similar triangles Sine and cosine ratios for similar triangles
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5.3 Apply the Sine and Cosine Ratios
Example 4 Sine and cosine ratios for similar triangles Sine and cosine ratios for similar triangles
Find the sine and cosine of , , , LYX and M of the
similar triangles. Then compare the ratios.
L
M
N a3
b3c3
X
Y
Z a
bc
In XYZ, and are _______________ angles, so sin X = cos ___ and sin Y = cos ___.
X Y complementaryY X
In LMN, and are _______________ angles, so sin L = cos ___ and sin M = cos ___.
L M complementaryM L
Because XYZ and LMN are _______ triangles, sin X = sin ___, cos X = cos ___, sin Y = sin ___, and cos Y = cos ___.
similarL L MM
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5.3 Apply the Sine and Cosine Ratios
Example 5 Use trigonometric ratios to find side lengths Use trigonometric ratios to find side lengths
Find the height of the parking ramp shown.
ft 65ft x
o27_______
o27sin hyp.
opp.
_______
o27sin 65
x
x o27sin___65
x______ ___65x_____5.29
4540.0
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5.3 Apply the Sine and Cosine RatiosCheckpoint. Complete the following exercises.Checkpoint. Complete the following exercises.
3. Find the value of x. Round to the nearest tenth.
28x
o46y
46sin o x
2828 28
x 28 7193.0x 1.20
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5.3 Apply the Sine and Cosine RatiosCheckpoint. Complete the following exercises.Checkpoint. Complete the following exercises.
4. Find the value of y. Round to the nearest tenth.
28x
o46y
46 cos o y
2828 28
y 28 6947.0y 5.19
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5.3 Apply the Sine and Cosine Ratios
Pg. 180, 5.3 #1-19