9.4 WarmupFind the measure of the missing leg in the

right triangle, and then calculate the ratio 𝒚𝟏

𝒙𝟏.

1. 2. 3. Solve

The two triangles are _________ so

the corresponding angles are ___________.

March 20, 2017 Geometry 9.5 Trigonometric Ratios 1

9

𝑦1 = 6

𝑥1

Geometry

9.4 & 9.5 Trigonometric Ratios

mbhaub@mpsaz.org

Essential Question

How is a right triangle used to find the

tangent of an acute angle? Is there a

unique right triangle that must be used?

March 20, 2017 Geometry 9.5 Trigonometric Ratios 3

March 20, 2017 Geometry 9.5 Trigonometric Ratios 4

What is triginometry?

It is the study of the relationships

between the sides and angles of

right triangles.

March 20, 2017 Geometry 9.5 Trigonometric Ratios 5

Trig Ratios

It is the ratio of the lengths of two sides

in a right triangle.

We will study only three:

Sine

Cosine

Tangent

March 20, 2017 Geometry 9.5 Trigonometric Ratios 6

Right Triangle

Leg

Leg

A

From A, this leg is the Adjacent side.

From A, this leg is

the Opposite side.

March 20, 2017 Geometry 9.5 Trigonometric Ratios 7

Right Triangle

Leg

Leg

A

From A, this leg is the Adjacent side.

From A, this leg is

the Opposite side.

B

From B, this leg is

the Adjacent side.

From B, this leg is the Opposite side.

March 20, 2017 Geometry 9.5 Trigonometric Ratios 8

Right Triangle

Adjacent

Opposite

A

March 20, 2017 Geometry 9.5 Trigonometric Ratios 9

Trig Ratio Definition: Tangent

Adjacent

Opposite

A

Tangent of A =OppositeAdjacent

March 20, 2017 Geometry 9.5 Trigonometric Ratios 10

Abbreviation

Tangent of A =OppositeAdjacenttan A

Opp

Adj

March 20, 2017 Geometry 9.5 Trigonometric Ratios 11

Writing Ratios SOH CAH TOA

3

4

5

A

B

tan 𝐴 =3

4tan𝐵 =

4

3

Example 1

Find tan R and tan S. Write each answer as

a fraction and as a decimal rounded to four

places.

March 20, 2017 Geometry 9.5 Trigonometric Ratios 12

tan R =

tan S =

18

80=

9

40= .225

80

18=

40

9= 4.4444

Your Turn

Find sin R, cos R, and tan R. Write each

answer as a fraction and as a decimal

rounded to four places.

March 20, 2017 Geometry 9.5 Trigonometric Ratios 13

tan A =

tan B =

15

18=

5

6= .8333

18

15=

6

5= 1.2

Using Special Right Triangles to find

tangent of a 30, 60 or 45 angle.

Use a special right triangle to

find the tangent of a 60° angle.

tan 60 =𝑜𝑝𝑝

𝑎𝑑𝑗

tan 60 =3

1

tan 60 = 3

March 20, 2017 Geometry 9.5 Trigonometric Ratios 14

x

x 3

tan 60 ≈ 1.7321

Using Special Right Triangles to find

tangent of a 30, 60 or 45 angle.

Use a special right triangle to

find the tangent of a 30° angle.

tan 30 =𝑜𝑝𝑝

𝑎𝑑𝑗

tan 30 =1

3

tan 30 =1

3∙

3

3March 20, 2017 Geometry 9.5 Trigonometric Ratios 15

x

x 3

tan 30 =3

3

tan 30 ≈ .5774

Using Special Right Triangles to find

tangent of a 30, 60 or 45 angle.

Use a special right triangle to

find the tangent of a 45° angle.

tan 45 =𝑜𝑝𝑝

𝑎𝑑𝑗

tan 45 =𝑥

𝑥

tan 45 = 1

March 20, 2017 Geometry 9.5 Trigonometric Ratios 16

March 20, 2017 Geometry 9.5 Trigonometric Ratios 17

Calculators

Make sure your calculator is in

DEGREE mode.

Always use four decimal places of

accuracy when using trig functions.

All demonstrations here are from a TI

graphing calculator.

March 20, 2017 Geometry 9.5 Trigonometric Ratios 18

Mode Setting

Press MODE

Use the cursor

arrows and move

to Degree.

Press ENTER.

Press 2nd Quit.

Press Clear

March 20, 2017 Geometry 9.5 Trigonometric Ratios 19

Using Trig Functions

To find the tan 78:

Press ‘tan’

Enter 78

Add a ).

Press ENTER.

Answer is 4.7046

March 20, 2017 Geometry 9.5 Trigonometric Ratios 20

Find these values:

tan 15

tan 60

tan 45

tan 80

.2679

1.7321

1

5.6713

March 20, 2017 Geometry 9.5 Trigonometric Ratios 21

Example 2 Find a.

tan 408

8tan 40

6.7

a

a

a

40

a

8

Write the equation and solve.

March 20, 2017 Geometry 9.5 Trigonometric Ratios 22

Example 3 Find a.

17tan 40

tan 40 17

17

tan 40

20.3

a

a

a

x

40

a

17

Write the equation and solve.

March 20, 2017 Geometry 9.5 Trigonometric Ratios 23

Your Turn Find x.

tan 78150

150 tan 78

705.7

x

x

x

78

x

150

150cos78

150

cos78

721.5

y

y

y

March 20, 2017 Geometry 9.5 Trigonometric Ratios 24

Angle of Elevation

Horizontal

Angle of

Elevation

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Angle of DepressionHorizontal

Angle of

Depression

March 20, 2017 Geometry 9.5 Trigonometric Ratios 26

Example 4

30 yd

15h

Standing 30 yards from a

tree, the angle of elevation

to the top of the tree is

15. How tall is the tree?

tan1530

30 tan15

8.0

h

h

h

March 20, 2017 Geometry 9.5 Trigonometric Ratios 27

Example 5

Isabella is 30 feet from a

fearsome monster. The angle

of elevation to the top of the

monster’s head is 42. How

tall is the monster?

30 ft42

x ft

March 20, 2017 Geometry 9.5 Trigonometric Ratios 28

Solution

30 ft42

x ft

tan 4230

30 tan 42

30(.9004)

27

x

x

March 20, 2017 Geometry 9.5 Trigonometric Ratios 29

Solution

30 ft42

27 ft

tan 4230

30 tan 42

30(.9004)

27

x

x

Your Turn

March 20, 2017 Geometry 9.5 Trigonometric Ratios 30

If the angle of elevation from your position on the

ground to the top of a building is 67° and you are

standing 30 meters from the foot of the building,

approximate the height of the building.

March 20, 2017 Geometry 9.5 Trigonometric Ratios 31

Homework