How do I use the sine, cosine, and tangent ratios to solve triangles?

• How do I use the sine, cosine, and tangent ratios to solve triangles?

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


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


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


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


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


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



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


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


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



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

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

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


Pg. 180, 5.3 #1-19

How do I use the sine, cosine, and tangent ratios to solve triangles?

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