Transcript 7-5 & 7-6 Sine, Cosine, and Tangent Ratios

Date: 2/7
Aim: 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
lengthof leg Opposite
Sine of  
lengthof Hypotenuse
lengthof leg Adjacent
Cosine of  
lengthof Hypotenuse
lengthof leg Opposite
T angentof  
lengthof Adjacent
SOH CAH TOA
is mnemonic for:
• SOH
Opposite
Sin 
Hypotenuse
• CAH
Adjacent
Cos 
Hypotenuse
• TOA
Opposite
T an 
Adjacent
Trigonometry
B
a
Sin A 
c
a
C
b
Cos A 
c
a
Tan A 
b
c
b
A
Using Trigonometry
•
Find thelengthsof 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
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
Date: 2/8
Aim: To use these ratios for indirect
measurements of right triangles.
Do Now: Find h.
Find the height h of the lamppost to the nearest inch.
opp.
tan 70o =
adj.
tan 70o =
h
40
40 tan 70o = h
109.9
h
ANSWER
The lamppost is about 110 inches tall.
• Mount Everest –
29, 029 feet
• The Burj Khalifa, Dubai – 2717 feet
• Angel, Salto – Venezuela - 3,212 feet
Clinometer
Angle
of
Elevation
42 °
42 °
Angle
of
Depression
• 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.
θ – angle of interest
Which angle is the angle of
elevation?
Which angle is the angle of
depression ?
A
A - angle of
elevation
θ
B
B – 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 Ms. Psillos from the mountain, if her
angle of elevation was 55°?
20,326.3 feet
Ms. Psillos is looking up at the waterfall in Angel,
Salto – Venezuela, which is 3,212 feet tall, and
has an angle of elevation of 82°. She wants to
know the distance that her line of sight makes
to the top of the waterfall?
3,243.6 feet
Using Trigonometry
• Find mR to thenearestdegree.
T angentRatio
47
Tan R 
41
R
41
Tan R  1.146
1
T an 1.146  mR
R  49
S
47
T