Right TrianglesPage 269

Objective

To solve right triangles.

To find the missing parts of right triangles using the trig functions.

Practical ApplicationThe longest truck mounted ladder used by the Dallas Fire Department is 108 feet long and consists of four hydraulic sections. Gerald Travis, aerial expert with the department, indicates that the optimum operating angle of this ladder is 60°. Outriggers, with an 18-foot span between each, are used to stabilize the ladder truck and permit operating angles greater than 60°, allowing the ladder truck to be closer to buildings in the downtown streets of Dallas. Assuming the ladder is mounted 8 feet off the ground, how far from an 84-foot burning building should the base of the ladder be placed to achieve the optimum operating angle of 60°? How far should the ladder be extended to reach the roof?

Right triangles can be used to define trig functions.

A

B

C

a

b

c

sin A = oppositehypotenuse

= ac

cos A = adjacenthypotenuse =

bc

tan A =oppositeadjacent =

ab

csc A = hypotenuse opposite

= ca

sec A = hypotenuse adjacent

= cb

cot A = adjacentopposite

=ba

SOH-CAH-TOA

This is a mnemonic device commonly used for remembering the first three equations

Sin OppositeHypotenuse

Cos AdjacentHypotenuse

Tan =OppositeAdjacent

A right triangle has sides whose lengths are 5 cm, 12 cm, and 13 cm. Find the values of the six trig functions of

5

12

13

sin 513

cos 1213

tan 512

csc 13 5

sec 1312

cot 12 5

Solve right triangle ABC. Round angle measures to the nearest degree and side measures to the nearest tenth.

A

BC

49°

b c

7

Since A and B are complementary 49° + B = 90°

B = 41°

41°

sin 49° = 7c

0.7547 ≈ 7c

c ≈ 9.3

tan 49° = 7b

1.1504 ≈7b

b ≈ 6.1

So, B = 41°, c = 9.3, and b = 6.1

Find the measure of angle R to the nearest degree.

S

T

R

8 14

sin R = oppositehypotenuse

814=

Use your calculator

8 ÷ 14 = 2nd SIN = 34.849905

R ≈ 35°

The application at the beginning of the lesson. Assume the ladder is mounted 8’ off the ground.

A. How far from the 84-foot burning building should the base of the ladder be placed to achieve the optimum operating angle of 60°?B. How far should the ladder be extended to reach the roof?

84’78’

d

l

tan 60° =

60°

76 d

1.7321 ≈76 d

d ≈ 43.9

sin 60° = 76 l

0.8660 ≈76 l

l ≈ 87.8

So the truck should 43.9 feet from the wall and the ladder should be extended 87.8 feet.

Angles of Elevation and Depression – page 272

• There are many other applications that require trigonometric solutions. Surveyors use special instruments to find the measures of angles of elevation and angles of depression. An angle of elevation is the angle between a horizontal line and the line of sight from an observer to an object at a higher level. An angle of depression is the angle between a horizontal line and the line of sight from the observer to an object at a lower level.

A flagpole 40’ high stands on top of the Wentworth Building. From a point P in front of Bailey’s drugstore, the angle of elevation of the top of the pole is 54°54’, and the angle of elevation of the bottom of the pole is 47°30’. To the nearest foot, what is the height of the building?

47°30’

54°54’

Let x be the height of the building and a be the distance from the point to the foot of the building.

x

a

tan 47°30’ = xa

tan 54°54’ = 40’40 + x a

Solve each equation for a. Then the following is true.

40 + x .tan 54°54’

= x .tan 47°30’

tan 47°30’(40 +x) = x (tan 54°54’)

40tan 47°30’ = x(tan54°54’ – tan 47°30’)

43.6523 = 0.331x

x = 131.68 ≈ 132 feet

Assignment

Page 273

# 15 - 24