Applications of Right Triangles

Section 1.5

Standard: MCC9-12.G.SRT.8

Essential Question: How do trig ratios apply to real world applications?

Hints on solving trigonometry applications or problems:

• If no diagram is given, draw one yourself.

• Mark the right angles in the diagram.

• Show the sizes of the other angles and the lengths of any segments that are known

• Mark the angles or sides you have to calculate.

Hints on solving trigonometry applications or problems-continued:

• Consider whether you need to create right triangles by drawing extra lines. For example, divide an isosceles triangle into two congruent right triangles.

• Decide whether you will need the Pythagorean Theorem, or a trig function: sine, cosine or tangent.

• Check that your answer is reasonable. The hypotenuse is the longest side in a right triangle.

angle of depression = angle of elevation

angle of depression

angle of elevation

Angle of Depression: the angle between the horizontal and the line of sight to an object below the horizontal.Angle of Elevation: the angle between the horizontal and the line of sight to an object above the horizontal.

(1). A forest ranger is on a fire lookout tower in a national forest.His observation position is 214.7 feet above the ground when he spots an illegal campfire. The angle of depression of the line of site to the campfire is 12°.

a. The angle of depression is equal to the corresponding angle of elevation. Why? b. Assuming that the ground is level, how far is it from the base of the tower to the campfire?

(1a). The angle of depression is equal to the corresponding angle of elevation. Why?

(1b). Assuming that the ground is level, how far is it from the base of the tower to the campfire?

214.

7Alternate Interior Angles are Congruent

x

x

7.214)12tan(

)12tan(

7.214x

1.1010x

The fire is 1,010.1 feet from the base of the tower.

(2). A ladder 5 m long, leaning against a vertical wall makes an angle of 65˚ with the ground. a) How high on the wall does the ladder reach? b) How far is the foot of the ladder from the wall? c) What angle does the ladder make with the wall?

65o

5 m

x

5)65sin(x

x )65sin(5

x5.4

The ladder reaches 4.5 m up the wall.

(2). A ladder 5 m long, leaning against a vertical wall makes an angle of 65˚ with the ground. b) How far is the foot of the ladder from the wall?

65o

5 m

y

5)65cos(y

y )65cos(5

y1.2

The foot of the ladder is 2.1 m from the wall.

(2). A ladder 5 m long, leaning against a vertical wall makes an angle of 65˚ with the ground. c) What angle does the ladder make with the wall?

65o

5 mzo

6590 z

The ladder makes a 25o angle with the wall.

25z

(3). The ends of a hay trough for feeding livestock have the shape of congruent isosceles trapezoids as shown in the figure below. The trough is 18 inches deep, its base is 30 inches wide, and the sides make an angle of 118° with the base. How much wider is the opening across the top of the trough than the base?

90o

28o

118o – 90o = 28o

18

x

18)28tan(

x

x )28tan(18

x6.9

The opening across the top is 2(9.6) or 19.2 in. wider than the base.

Investigation 1: In your group, select one person as the recorder, one as the calculation expert, and one as the presenter. Work the problem assigned to your group on poster paper including a labeled diagram, equation with solution, and answer written in sentence form. The presenter will demonstrate your solution to the class.

(4). A guy wire is attached from the top of a tower to a point 80m from the base of the tower. (a). If the angle of elevation to the top of the tower of the wire is 28°, how long is the guy wire? (b). How tall is the tower?

28o

80 m

x

x

80)28cos(

x6.90y

80)28tan(

y

y )28tan(80

y5.42

(4a). (4b).

The wire is 90.6 m long and thetower is 42.5 m tall.

)28cos(

80x

(5). How tall is a bridge if a 6-foot-tall person standing 100 feet away can see the top of the bridge at an angle of elevation of 30° to the horizon?

30o

100 ft

100 ft

x

6 ft

100)30tan(

x

x )30tan(100

x7.57

The bridgeis 57.7 + 6 or 63.7 ft tall

(6). A bow hunter is perched in a tree 15 feet off the ground. The angle of depression of the line of site to his prey on the ground is 30o. How far will the arrow have to travel to hit his target?

15

30o

30o

x

x

15)30sin(

)30sin(

15x

30x

The arrow willtravel 30 ft toits target.

(7). Standing across the street 50 feet from a building, the angle to the top of the building is 40°. An antenna sits on the front edge of the roof of the building. The angle to the top of the antenna is 52°. a) How tall is the building?

40o

50 ft

x

50)40tan(

x

x )40tan(50

x0.42

The building is 42 ft tall.

(7). Standing across the street 50 feet from a building, the angle to the top of the building is 40°. An antenna sits on the front edge of the roof of the building. The angle to the top of the antenna is 52°. b) How tall is the antenna itself, not includingthe height of the building?

52o

50 ft

y

50)52tan(

y

y )52tan(50

y0.64

The antenna is 64.0 – 42.0 or 22 feet tall.

(8). An air force pilot must descent 1500 feet over a distance of 9000 feet to land smoothly on an aircraft carrier. What is the plane’s angle of descent?

9000 ft

1500 ftxo

xo

9000

1500)tan( x

9000

1500tan 1x

5.9x

The angle of descent is approximately 9.5o.

Homework #1-7at bottom of Note Handout1.5 Applications of Right Triangles