Goal: To use trigonometric ratios for indirect measurements of right triangles.

7.6 Apply Sine and Cosine Ratios

TrigonometryTrigonometry• Derived from Greek words, Derived from Greek words, “trigonon"“trigonon" which means which means

triangle, and triangle, and "metria”"metria” which means measure. which means measure.

• Trigonon metriaTrigonon metria = = “triangle measurement” “triangle measurement”

• Sine - comes from the Latin word sinus meaning a curve or fold.

• Cosine - was originally written "co.sine," short for COMPLEMENTI SINUS: the sine of the complementary angle.

• Tangent - comes from the Latin word tangens which is a straight line which touches a circle at one point.

Triangle Ratios

Hypotenuse oflength

Opposite leg oflength of Sine

Hypotenuse oflength

Adjacent leg oflength of Cosine

Adjacent oflength

Opposite leg oflength ofTangent

SOH CAH TOAis mnemonic for:

• SOH

• CAH

• TOA

Hypotenuse

OppositeSin

Hypotenuse

AdjacentCos

Adjacent

OppositeTan

Trigonometry

C A

B

a

b

c

c

aA Sin

c

bA Cos

b

aA Tan

42 °

Angle of

Depression

Angle of

Elevation

42 °

You want to build a skateboard ramp with a length of 14 feet and an angle of elevation of 26°. You need to find the height and length of the base of the ramp.

sin 26o=

opp. hyp.

sin 26o x

=14

14 sin 26o = x

6.1 ft. x

cos 26o=

adj. hyp.

cos 26o y

=14

14 cos 26o = y

12.6 ft. y

Using Trigonometry

B

CA29

. and of lengths theFind ca

c

2937Sin

73Sin

29c

3.30c

•

ca

73

a

2937Tan

73Tan

29a

9.8a

tan 32o=

opp. adj.

tan 32o 11

=x

x tan 32o = 11

x = 11 tan 32o

x 11 0.6249

x 17.6

Try on your own:

y

1132Sin

23Sin

11y

8.20y

y

• A 20-ft wire supporting a flagpole forms a 35o angle with the flagpole. To the nearest foot, how high is the flagpole?

20

35

x

Ratio Cosine

2035 Cos

x

)35 Cos(20 x

ft 16x

Using Trigonometry

• Mount Everest – 29, 029 feet

• The Burj Khalifa, Dubai – 2717 feet

• Angel, Salto – Venezuela - 3,212 feet

Clinometer

• The angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (the line of sight).

• If the object is below the level of the observer, then the angle between the horizontal and the observer's line of sight is called the angle of depression.

Exact Answer:1,250 feet tall

How tall is the Empire State Building?

76.5°

300 ft

Since Mount Everest is 29, 029 feet tall, how far was Mr. Sharma from the mountain, if his angle of elevation was 55°?

20,326.3 feet

Mr. Sharma is looking up at the waterfall in Angel, Salto – Venezuela, which is 3,212 feet tall, and has an angle of elevation of 82°. He wants to know the distance that his line of sight makes to the top of the waterfall?

3,243.6 feet

RatioTangent

Using Trigonometry

S T

R

41

47

degree.nearest the to Find Rm

41

47Tan R

146.1Tan R

Rm 146.1Tan 1

49R

•