9.4/9.5 Warmup

Find the measure of the missing leg in the

right triangle, and then calculate the ratio 𝒚𝟏

𝒙𝟏.

1. 2.

The two triangles are _____________ so two

angles in each triangle are ___________.

March 28, 2016 Geometry 9.5 Trigonometric Ratios 1

9

Geometry

9.4 & 9.5 Trigonometric Ratios

mbhaub@mpsaz.org

Essential Question

How is a right triangle used to find the sine,

cosine, and tangent of an acute angle?

March 28, 2016 Geometry 9.5 Trigonometric Ratios 3

March 28, 2016 Geometry 9.5 Trigonometric Ratios 4

Goals

Find the sine, cosine, and tangent of

an acute angle.

Solve problems using trigonometric

ratios.

March 28, 2016 Geometry 9.5 Trigonometric Ratios 5

Terminology

No one, except stuffed-shirt

mathematics teachers, uses the

word trigonometry.

It’s

March 28, 2016 Geometry 9.5 Trigonometric Ratios 6

What is trig?

Literally, the measure of triangles.

An extremely useful, practical and

powerful math tool.

A branch of math that finds its way into

practically everything we do.

Usually learned in high school.

March 28, 2016 Geometry 9.5 Trigonometric Ratios 7

What you will learn…

The basic terms and methods of

solving right triangles.

How to use a calculator’s trig

functions.

How to solve problems using trig.

March 28, 2016 Geometry 9.5 Trigonometric Ratios 8

Trig Ratios

Based on the sides of a right triangle.

We will study only three:

Sine

Cosine

Tangent

March 28, 2016 Geometry 9.5 Trigonometric Ratios 9

Right Triangle

Leg

Leg

A

From A, this leg is the Adjacent side.

From A, this leg is

the Opposite side.

March 28, 2016 Geometry 9.5 Trigonometric Ratios 10

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 28, 2016 Geometry 9.5 Trigonometric Ratios 11

Right Triangle

Adjacent

Opposite

A

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Trig Ratio Definition: Sine

Adjacent

Opposite

A

Sine of A =Opposite

Hypotenuse

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Trig Ratio Definition: Cosine

Adjacent

Opposite

A

Cosine of A =Adjacent

Hypotenuse

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Trig Ratio Definition: Tangent

Adjacent

Opposite

A

Tangent of A =OppositeAdjacent

March 28, 2016 Geometry 9.5 Trigonometric Ratios 15

Abbreviations

Tangent of A =OppositeAdjacent

Sine of A =Opposite

Hypotenuse

Cosine of A =Adjacent

Hypotenuse

sin A

cos A

tan A

March 28, 2016 Geometry 9.5 Trigonometric Ratios 16

Memory Aid

Sine is Opposite over Hypotenuse.

Cosine is Adjacent over Hypotenuse.

Tangent is Opposite over Adjacent.

SOH CAH TOA

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Trig RatiosA

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Writing Ratios SOH CAH TOA

4sin

5

3cos

5

4tan

3

B

B

B

3

4

5

A

B 3sin

5

4cos

5

3tan

4

A

A

A

?

?

?

?

?

?

?

?

?

?

?

?

March 28, 2016 Geometry 9.5 Trigonometric Ratios 19

Writing Ratios SOH CAH TOA

4sin

5

3cos

5

4tan

3

B

B

B

3

4

5

A

B 3sin

5

4cos

5

3tan

4

A

A

A

Example 1

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

answer as a fraction and as a decimal

rounded to four places.

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sin S =

cos S =

tan S =

80

82=

40

41= .9756

18

82=

9

41= .2195

80

18=

40

9= 4.444

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 28, 2016 Geometry 9.5 Trigonometric Ratios 21

sin R =

cos R =

tan R =

80

82=

40

41= .9756

18

82=

9

41= .2195

18

80=

9

40= .2250

March 28, 2016 Geometry 9.5 Trigonometric Ratios 22

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.

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Mode Setting

Press MODE

Use the cursor

arrows and move

to Degree.

Press ENTER.

Press 2nd Quit.

Press Clear

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Using Trig Functions

To find the sin 78:

Press ‘sin’

Enter 78

Press ENTER.

Answer is .9781

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Find these values:

sin 15

cos 45

tan 45

cos 80

sin 10

tan 5

cos 60

sin 90

.2588

.7071

1

.1736

.1736

.0875

.5

1

March 28, 2016 Geometry 9.5 Trigonometric Ratios 26

Solving Triangles

Carefully analyze the given

information.

Decide what you are trying to find.

Ask: Which trig function fits this

problem?

WRITE AN EQUATION. (SOH CAH TOA)

Solve.

March 28, 2016 Geometry 9.5 Trigonometric Ratios 27

Example 2 Find x.

28

x 15

From the 28 angle, x is the ?

Opposite side,

and 15 is the

Hypotenuse.

What trig ratio is this?

Sine (SOH CAH TOA)

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Example 2 Find x.

sin 2815

15sin 28

7.0

x

x

x

28

x 15

Write the equation and solve.

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Example 3 Find y.

cos3156

56cos31

48.0

y

y

y

31

y

56

Write the equation and solve.

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Example 4 Find a.

tan 408

8tan 40

6.7

a

a

a

40

a

8

Write the equation and solve.

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Fraction Reminder

bc a

82

4

ba c

84

2

If Then

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Example 5 Find a.

17tan 40

tan 40 17

17

tan 40

20.3

a

a

a

x

40

a

17

Write the equation and solve.

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Example 6 Find x & y.

tan 78150

150 tan 78

705.7

x

x

x

78

x y

150

150cos78

150

cos78

721.5

y

y

y

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Angle of Elevation

Horizontal

Angle of

Elevation

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Angle of DepressionHorizontal

Angle of

Depression

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Example 7

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 28, 2016 Geometry 9.5 Trigonometric Ratios 37

Example 8

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

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Solution

30 ft42

x ft

tan 4230

30 tan 42

30(.9004)

27

x

x

March 28, 2016 Geometry 9.5 Trigonometric Ratios 39

Solution

30 ft42

27 ft

tan 4230

30 tan 42

30(.9004)

27

x

x

Your Turn

You are skiing on a mountain. You start at an altitude

of 8400 feet and ski down to an altitude of 7200. The

angle of depression is 21°. Find the distance x you ski

down the mountain to the nearest foot.

March 28, 2016 Geometry 9.5 Trigonometric Ratios 40

y

You ski about 3349 ft down the

mountain.

March 28, 2016 Geometry 9.5 Trigonometric Ratios 41

Summary

Trig ratios are based on acute angles

in right triangles.

They are Sine, Cosine, Tangent.

SOH CAH TOA

Angle of elevation is from the ground

up.

March 28, 2016 Geometry 9.5 Trigonometric Ratios 42

Homework