CHAPTER 9.10

TRIGONOMETRIC TRIGONOMETRIC RATIOSRATIOS

By: Arielle GreenMod 9

Sin =

Cos =

Tan =

hypotenuse

opposite

hypotenuse

adjacent

adjacent

opposite

VOCABULARY Angle of Elevation – the angle between an

upward line of sight and the horizontal

is the angle of elevation.

Horizontal line

Line of sight

C

A

ABC

B

angle of elevation

SAMPLE PROBLEM 1 A girl was walking in the woods when she

stopped 10 ft away from a tree. She spotted a birds nest at an angle of elevation of 37˚. How far up from the ground was the birds nest rounded to the nearest tenth?

10

37˚

X

First choose the formula needed for this problem. We are working with the two legs of the right triangle, so we will use tan. Set up the formula and solve for x.

5.7

37tan1010

37tan

x

x

x

R S

ft

Q

VOCABULARY

X

YZ

line of sight

horizontal line

W

WXY

Angle of depression

An airplane pilot is flying over a forest at an altitude of 1600 ft. Suddenly, he spots a fire. He measures the angle of depression and finds it to be 46˚. How far is the fire, rounded to the nearest tenth, from a point on land directly below the plane?

There are two ways to solve this problem. We’ll look at both ways.

1600

X

A

B C

46˚

46˚

Using parallel lines alt. int. , <ACB is also 46˚. Since only the two legs of the right triangle are being used, the formula must be Tan= .

s

x

1600

adjacent

opposite

Set up the equation and solve for x. Tan 46 =

x =

x 1545.1 ft

D

46tan

1600

A

B C

1600

46˚44˚

X

D

Since <BAC and <CAD are complementary <s, <BAC is 44˚. Only the two legs of the right triangle are being used, so the formula must be Tan = .

adjacent

opposite

Set up the equation and solve for x. Tan 44 =

X = 1600 ∙ tan 44

x 1545.1 ft

1600

x

PRACTICE PROBLEMS ROUND ALL ANSWERS TO THE NEAREST TENTH. ROUND ALL ANGLES TO THE NEAREST DEGREE.

1.) A lighthouse casts a shadow of 55 ft when the sun is at an angle of elevation of 67˚. How tall is the lighthouse?

2.) A cat was on a cliff when it saw a mouse down below at an angle of depression of 25˚. The cliff is 43 ft tall. How far away is the mouse from the bottom of the cliff?

3.)A 25-foot ladder just reaches a point on a wall 24 ft above the ground. What is the angle of elevation of the ladder?

PRACTICE PROBLEMS

Two men are on the opposite sides of a tall building with the angle of elevation being 30 and 60 respectively. If the one man is 40 feet away from the base of the building, how far away is the other man?

A pole 40 ft high has a shadow the length of 23 ft at this point in time. Find the angle of elevation of the sun.

Harry was walking along a pier. He stopped when he saw a boat on the lake at an angle of depression of 22˚. If the boat is 65 ft away, how high, rounded to the nearest tenth, is the pier from the water ?

4.) 5.)

6.)

x30˚ 60˚

40

ANSWERS TO THE PRACTICE PROBLEMS

1.)

2.)

3.)

x

5567˚

ftx

x

x

6.129

67tan5555

67tan

43

65˚

x ftx

x

x

2.92

65tan4343

65tan

X˚

2425

7.73

25

24sin

25

24sin

x

x

x

¯¹

ANSWERS TO PRACTICE PROBLEMS (CONT’D)

4.) 5.)

8

60˚30˚

x40

094.23

30tan4040

30tan

y

y

y

3.1360tan

094.23

094.2360tan

x

x

x

23

.09

4

ft

40

23

x˚¯¹

1.60

23

40tan

23

40tan

x

x

x

˚

6.)

68˚

65

x

3.2668tan

65

6568tan

x

x

x

ft22˚

22˚

WORKS CITED

Rhoad,Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and

Challenge. Boston: McDougal Little, 2004. 423-427.

“Math:Trigonometry.”Syvum. 2008. Syvum technologies. 29 May 2008. <

http://www.syvum.com/cgi/online/serve.cgi/mat h/trigo/trig3.sal >.