Transcript 7.6-Sine-cosineC

7.6 Apply Sine and Cosine Ratios
Goal: To use trigonometric ratios for indirect
measurements of right triangles.
Trigonometry
• Derived from Greek words, “trigonon" which means
triangle, and "metria” which means measure.
•
Trigonon metria = “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
length of leg Opposite 
Sine of  
length of Hypotenuse
length of leg Adjacent 
Cosine of  
length of Hypotenuse
length of leg Opposite 
Tangent of  
length of Adjacent 
SOH CAH TOA
is mnemonic for:
• SOH
Opposite
Sin 
Hypotenuse
• CAH
Adjacent
Cos 
Hypotenuse
• TOA
Opposite
Tan 
Adjacent
Trigonometry
B
a
Sin A 
c
a
C
b
Cos A 
c
a
Tan A 
b
c
b
A
EXAMPLE 3 -HOTS
THINK….INK….SHARE
EXAMPLE 3 SOLUTIONS
Angle
of
Depression
42 °
42 °
Angle
of
Elevation
EXAMPLE 4 – HOTS FINDING A
HYPOTENUSE USING AN ANGLE OF
DEPRESSION
EXAMPLE 4 SOLUTION
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.
adj.
cos 26o =
hyp.
opp.
sin 26o =
hyp.
sin 26o =
x
14
14 sin 26o = x
6.1 ft.
x
cos 26o =
y
14
14 cos 26o = y
12.6 ft.
y
Using Trigonometry
•
Find the lengths of a and c.
29
Sin 73 
c
29
c
Sin 73
c  30.3
29
A
c
29
Tan 73 
a
29
a
Tan 73
a  8.9
C
73
a
B
Try on your own:
y
opp.
tan 32o =
adj.
tan 32o = 11
x
x tan 32o = 11
11
x =
tan 32o
11
x
0.6249
x 17.6
11
Sin 32 
y
11
y
Sin 32
y  20.8
Using Trigonometry
• A 20-ft wire supporting a flagpole forms a 35o
angle with the flagpole. To the nearest foot,
how high is the flagpole?
Cosine Ratio
x
Cos 35 
20
x  20(Cos 35)
x  16 ft
35
20
x
• 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.
How tall is the Empire State Building?
Exact Answer:
1,250 feet tall
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
Using Trigonometry
• Find mR to the nearest degree.
Tangent Ratio
47
Tan R 
41
R
41
Tan R  1.146
1
Tan 1.146  mR
R  49
S
47
T