Chapter 12 - Lesson 12.2Problem Solving with Right Triangles

HW: 12.2/1-20

Finding the Sides of a Triangle

Remember: SOHCAHTOA

A

OT

H

AC

H

OS

Review: Trig Ratios

Sin P

Cos P

1220

16Q

P

Tan P Tan Q

Cos Q

Sin Q16

20

12

20

16

12

12

20

16

20

12

16

First we will find the Sine, Cosine and Tangent ratios for Angle P.

Next we will find the Sine, Cosine, and Tangent ratios for Angle Q

Opposite

Adjacent

Remember SohCahToa

Solving Right Triangles

Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs.

To Solve a Right Triangle means to determine the measures of all six parts.

Missing sides r and s, and angle S.

70

2090

S

S

1520sin

r

r20sin15

1.5

1303.5

r

r

1520cos

s

s20cos15

1.14

0954.14

s

s

But what if you don’t know either of the acute angles?To solve those triangle we must use

Inverse Trig Functions

25.18

10tan A

25.1tan 1 A 3.51A

7.383.5190B222 108 b

8.12164

10064

b

b

6.66332.6

44100144

1012 222

b

b

bMissing side….

6

5

12

10cos B

9.36869.36

8.0cos

8.0cos1

B

B

B

1.53

9.3690

Am

Am

What are the angles of elevation and depression and what is their

relationship to right triangles?

ANGLE OF DEPRESSION

cliffs

Eye level

Angle of depression

Sea level

object

observer

Looking down from the horizontal

cliffs

Eye level

Angle of elevation

Sea level

object

observer

ANGLE OF ELEVATION

Looking up from the horizontal

If an observer sights an object above, the angle between a horizontal line and his or

her line of sight is called an angle of elevation. If the line of sight is below the

horizontal it is called the angle of

depression.

Angle of Elevation

Angle of Depression

eye - level

eye - level

line of

line of

sight

sight

eye - level

eye - level

line of

sight

eye - level

eye - level

The angles are equal – they are alternate interior angles

line of

sight

Angles of Elevation and Depression

Since the two horizontal lines are parallel, by Alternate Interior Angles the angle of depression must be equal to the angle of elevation.

Bottom Horizontal

Top Horizontal

Line of Sight

Angle of Elevation

Angle of Depression

Angles of Elevation and Depression

line of sight

B

A21 m

h

Example 1

The angle of elevation of building A to building B is 250. The distance between the buildings is 21 meters. Calculate how much

taller Building B is than building A.

Step 1: Draw a right angled triangle with the given information.

Step 3: Set up the trig equation. mh

h

8.9

25tan21

Angle of elevation

Step 4: Solve the trig equation.

2125tan

h

250

Step 2: Take care with placement of the angle of elevation

Step 1: Draw a right angled triangle with the given information.

Step 3: Decide which trig ratio to use.

60 m

80 m

60

80tan

Step 4: Use calculator to find the value of the unknown. 53.1º

A boat is 60 meters out to sea. Madge is standing on a cliff 80 meters high. What is the angle of depression

from the top of the cliff to the boat?

Step 2: Alternate interior angles place inside the triangle.

Example 2

Angle of depression

60

80tan 1

Step 1: Draw a right angled triangle with the given information.

Step 3: Decide which trig ratio to use.

2520tan 0 h

Step 4: Use calculator to find the value of the unknown.

Step 2: Alternate interior angles places 200 inside the triangle.

(nearest km)

Example 3

Marty is standing on level ground when he sees a plane directly overhead. The angle of elevation of the plane after it has travelled

25 km is 200. Calculate the altitude of the plane at this time.

200

h

25 km 200

Plane

Angle of elevation

Examples

Example 4

Kate and Petra are on opposite sides of a tree. The angle of elevation to the top of the tree from Kate is 45o and from Petra is

65o. If the tree is 5 m tall, who is closer to the tree, Kate or Petra?

K P450 650

5m

k p

k

545tan

Kate

p

565tan

Petra

Therefore, Petra is closer to the tree, since the

distance is shorter.

Answer

mk

k

5

45tan

5

).2(3.2

65tan

5

figssigmp

p

Example 5

Maryann is peering outside her window. From her window she sees her car and a bird hovering above her car. The angle of

depression of Maryann’s car is 200 whilst the angle of elevation to the bird is 400. If Maryann’s window is 2m off the ground, what is

the bird’s altitude at that moment?Step 1: Draw a diagram

Step 2: Set up the trig equations in two parts. Find d first, then b.

Step 3: Solve the equations and answer the question.

Bird

Car

400

200

2 m

bd

md

d

d

5.5

20tan

2

220tan

mb

b

b

6.4

40tan5.5

5.540tan

Therefore,

The bird is 6.6 m (2 + 4.6) from the ground at that moment.

Your Turn 1:

You sight a rock climber on a cliff at a 32o angle of elevation. The horizontal ground distance to the cliff is

1000 ft. Find the line of sight distance to the rock climber.

32

1000 ft

x

x

100032 Cos

32 Cos

1000x

ft 1179x

An airplane pilots sights a life raft at a 26o angle of depression. The airplane’s altitude is 3 km. What is

the airplane’s surface distance d from the raft?

26

26

3 km

d

d

326Tan

26Tan

3d

km 2.6d

Your Turn 2:

FYI: Theodolite

Theodolites are still used today for ultra

high precision optical alignment and measurement

Theodolites are mainly used for surveying

applications, and have been adapted for

specialized purposes in fields like meteorology

and rocket launch technology.

When the telescope is pointed at a target

object, the angle of each of these axes can be measured with great

precision

Theodolite is a precision instrument for measuring angles in the horizontal and vertical planes.

A surveyor stands 200 ft from a building to measure its height with a 5-ft tall theodolite. The angle of elevation to the top of the

building is 35°. How tall is the building?

x = 200 • tan 35°

tan 35° = x 200

So x ≈ 140

To find the height of the building, add the height of the Theodolite, which is 5 ft tall.

The building is about 140 ft + 5 ft, or 145 ft tall.

Your Turn 3:

An airplane flying 3500 ft above ground begins a 2° descent to

land at an airport. How many miles from the airport is the airplane

when it starts its descent?

sin 2° = 3500x ft

x ft = 3500 ft sin 2°

The airplane is about 19 mi from the airport when it starts its descent.

Your Turn 4:

x ft ≈ 100,287.9792 ft

mileftft /52809792.100287

x miles ≈ 18.9939 miles ≈ 19 miles

A 6-ft man stands 12 ft from the base of a tree. The angle of

elevation from his eyes to the top of the tree is 40°.

1.About how tall is the tree?

2. If the man releases a pigeon that flies directly to the top of the tree, about how far will it fly?

3.What is the angle of depression from the treetop to the man’s eyes?

about 16 ft

about 15.7 ft

40°

Your Turn 5:

CIRCUS ACTS. At the circus, a person in the audience watches the high-wire routine. A 5-foot-6-inch tall acrobat

is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience

member’s line of sight to the top of the acrobat is 27°?

27°

Step 1: Draw a triangle to fit problem

x

30.5 = 25 + 5.5Step 2: Label sides from angle’s view

adj

oppStep 3: Identify trig function to use

S O / HC A / HT O / A

Step 4: Set up equation

30.5tan 27° = ------- x

Step 5: Solve for variable

x tan 27° = 30.5 x = (30.5) / (tan 27°) x = 59.9

Your Turn 6:

DIVING At a diving competition, a 6-foot-tall diver stands atop the 32-foot platform. The front edge of the platform

projects 5 feet beyond the ends of the pool. The pool itself is 50 feet in length. A camera is set up at the

opposite end of the pool even with the pool’s edge. If the camera is angled so that its line of sight extends to the top of the diver’s head, what is the camera’s angle of

elevation to the nearest degree?

37 = 32 + 6

45 = 50 - 5

x°

Your Turn 7:

45

37tan x

45

37tan 1x

39427.39x

From a point 80m from the base of a tower, the angle of elevation is 28˚. How tall is the tower to the nearest

meter?

80m

28˚

x

tan 28˚ = x

8080 (tan 28˚) = x

80 (.5317) = x

x ≈ 42.5m is the height of the tower

Your Turn 8:

A ladder that is 20 ft is leaning against the side of a building. If the angle formed between the ladder and

ground is 75˚, how far is the bottom of the ladder from the base of the building?

ladd

er

bu

ildin

g20

x

75˚

cos 75˚ = x

20

20 (cos 75˚) = x

20 (.2588) = x

x ≈ 5.2ft from the base of the building

Your Turn 8:

When the sun is 62˚ above the horizon, a building casts a shadow 18m long. How tall is the building?

x

18shadow

62˚

tan 62˚ = x18

18 (tan 62˚) = x

18 (1.8807) = xx ≈ 33.9m is the height of the building

Your Turn 9:

A kite is flying at an angle of elevation of about 55˚. Ignoring the sag in the string, find the height of the kite if

85m of string have been let out.

string

85

x

55˚

kite

sin 55˚ = x

8585 (sin 55˚) = x85 (.8192) = x

x ≈ 69.6m is the height of the kite

Your Turn 10:

A 5.5 foot person standing 10 feet from a street light casts a 14 foot shadow. What is the height of the

streetlight?

5.5

14 shadowx˚

tan x˚ = 5.5

14x° ≈ 21.45°

tan 21.4524

height

10

Your Turn 11:

1st find the angle of elevation 2nd use the angle to find the height of the light.

45.21tan24height

height = 9.43 feet

The angle of depression from the top of a tower to a boulder on the ground is 38º. If the tower is 25m high,

how far from the base of the tower is the boulder?

25

x

angle of depression38º

38º

Alternate Interior Angles are congruent

Your Turn 12:

x

2538tan

38tan

25x

mx 3299.31