Jeff Bivin -- LZHS Right Triangle Trigonometry By: Jeffrey Bivin Lake Zurich High School...
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Jeff Bivin -- LZHS Right Triangle Trigonometry By: Jeffrey Bivin Lake Zurich High School [email protected] Last Updated: December 1, 2010
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Transcript of Jeff Bivin -- LZHS Right Triangle Trigonometry By: Jeffrey Bivin Lake Zurich High School...
Jeff Bivin -- LZHS
Right Triangle Trigonometry
By: Jeffrey BivinLake Zurich High School
Last Updated: December 1, 2010
Jeff Bivin -- LZHS
SOH CAH TOA
θ
oppositehypotenuse
adjacent
hypotenuse
oppositesin
hypotenuse
adjacentcos
adjacent
oppositetan
Jeff Bivin -- LZHS
Reciprocal Identities
θ
oppositehypotenuse
adjacent
hypotenuse
oppositesin
hypotenuse
adjacentcos
adjacent
oppositetan
sin
1csc
cos
1sec
tan
1cot
opposite
hypotenuse
adjacent
hypotensue
opposite
adjacent
Jeff Bivin -- LZHS
a
Find the sides.
1
2
130sin 0
o30A C
B
b
c o602
3
2
330cos 0
3
3
3
130tan 0
2
360sin 0
2
160cos 0
31
360tan 0
Jeff Bivin -- LZHS
a
Find the sides.
10
c
1030sin 0
b
1030tan 0
b
10
3
3
20c
1033 b
o30A C
B
b
c
102c
c
10
2
1
3103
30b
o60
Jeff Bivin -- LZHS
a
Find the sides.
17
c
1760cos 0
1760tan 0 b
173
b
34c
b317
o30A C
B
b
c
172c
c
17
2
1
o60
Jeff Bivin -- LZHS
a
Find the sides.
c
945cos 0
945tan 0 a
91
a
292
18c
a9
o45A C
B
b
c
922 c
c
9
2
2
o45
9
Jeff Bivin -- LZHS
Find the sides.
22o
1522sin
ao
68oc
A C
B
b
15a
ao 22sin15
a619.5
1522cos
bo
bo 22cos15
a908.13
Jeff Bivin -- LZHS
Find the sides.
34o
co 25
34sin
56oc
A C
B
b
2534sin oc
707.44c
a25
oc
34sin
25
bo 25
34tan
2534tan ob
064.37b
ob
34tan
25
Jeff Bivin -- LZHS
Find the angles and the 3rd side.
θ
25
21
25
21cos
)(cos 25211
25
21sin
)(sin 25211
o860.32
β
o140.57
441625 184 462
Jeff Bivin -- LZHS
Find the angles and the 3rd side.
θ
6
11
11
6tan
)(tan 1161
6
11tan
)(tan 6111
o610.28
β
o390.61
36121 157
157
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Quotient Identities
θ
oppositehypotenuse
adjacent
sintan
cos
cos
cotsin
hypotenuse
oppositesin
hypotenuse
adjacentcos
adjacent
oppositetan
1cot
tan
adjacent
opposite
Jeff Bivin -- LZHS
Pythagorean Identities
θ
1cossin 22 22 sec1tan
oppositehypotenuse
adjacent
222 hypadjopp
2
2
2
2
2
2
hyp
hyp
hyp
adj
hyp
opp
222 hypadjopp
2
2
2
2
2
2
adj
hyp
adj
adj
adj
opp
22 csccot1
222 hypadjopp
2
2
2
2
2
2
opp
hyp
opp
adj
opp
opp
Jeff Bivin -- LZHS
Using Pythagorean Identities
Find cosθ if sinθ = 9
5
1cossin 22
1cos9
5 22
81
56
81
251cos2
56cos
81
2 14cos
9
Jeff Bivin -- LZHS
Using Pythagorean Identities
Find secθ if tanθ = 8
7
22
sec18
7
2sec164
49
64
113
22 sec1tan
113sec
8
Jeff Bivin -- LZHS
Using Pythagorean Identities
Find sinθ if cotθ = 3
17
22
csc3
171
2csc19
289
9
298
298csc
3
22 csccot1
3 298 3sin
298 298
Jeff Bivin -- LZHS
θ cos(θ) sin(θ)
10o 0.985 0.174
20o 0.939 0.342
30o 0.866 0.5
35o 0.819 0.574
40o 0.766 0.643
50o 0.643 0.766
55o 0.574 0.819
60o 0.5 0.866
70o 0.342 0.939
80o 0.174 0.985
Co-function IdentitiesUse your calculators to evaluate each of the following.
Co
mp
limen
tary
An
gle
s
cos(θ) = sin(90o – θ) and sin(θ) = cos(90o – θ)
Jeff Bivin -- LZHS
Co-function IdentitiesC
om
plim
enta
ry A
ng
les
cos(θ) = sin(90o – θ) and sin(θ) = cos(90o – θ)
sec(θ) = csc(90o – θ) and csc(θ) = sec(90o – θ)
1 1sec csc 90
cos sin 90o
o
1 1csc sec 90
sin cos 90o
o
Jeff Bivin -- LZHS
Co-function IdentitiesC
om
plim
enta
ry A
ng
les
cos(θ) = sin(90o – θ) and sin(θ) = cos(90o – θ)
sec(θ) = csc(90o – θ) and csc(θ) = sec(90o – θ)
cos 90sintan cot 90
cos sin 90
o
o
o
sin 90coscot tan 90
sin cos 90
o
o
o
tan(θ) = cot(90o – θ) and cot(θ) = tan(90o – θ)
Jeff Bivin -- LZHS
Co-function IdentitiesC
om
plim
enta
ry A
ng
les
cos(θ) = sin(90o – θ) and sin(θ) = cos(90o – θ)
sec(θ) = csc(90o – θ) and csc(θ) = sec(90o – θ)
tan(θ) = cot(90o – θ) and cot(θ) = tan(90o – θ)
θ
90o- θ
a
b
c
cos sin 90ob
c
sin cos 90oa
c
tan cot 90oa
b
Jeff Bivin -- LZHS
(a, b)
(a, -b)
(b, a)
t
-t
t
90o-t
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
A surveyor is standing 45 feet from the base of a large tree. The surveyor measures the angle of elevation from the ground to the top of the tree to be 67.5o. Find the height of the tree.
45 feet
67.5o
htan(67.5 )
45o h
45 tan(67.5 )o h
108.640 h
108.640 .ft
Jeff Bivin -- LZHS
An airplane flying at 4500 feet is on a flight path directly toward an observer. If 30o is the angle of elevation from the observer to the plane, find the distance from the observer to the plane.
4500
fee
t30o
d
4500sin(30 )o
d
sin(30 ) 4500od
12 4500d
9000 .ft
9000d
Jeff Bivin -- LZHS
In traveling across flat land a driver noticed a mountain directly in front of the car. The angle of elevation to the peek is 4o. After the driver traveled 10 miles, the angle of elevation was 11o. Approximate the height of the mountain.
10 mi
h
tan(4 )10
o h
x
1.092 mi
11o
x
10 + x
tan(11 )oh
x
tan(11 )ox h
(10 ) tan(4 )ox h
(10 ) tan(4 ) tan(11 )o ox x 10 tan(4 ) tan(4 ) tan(11 )o o ox x
10 tan(4 ) tan(11 ) tan(4 )o o ox
10tan(4 )
tan(11 ) tan(4 )
o
o o x
10tan(4 )
tan(11 ) tan(4 )tan(11 )
o
o o
o h
4o
Jeff Bivin -- LZHS
A flagpole at the top of a tall building (and at the edge of the building) is know to be 45 feet tall. If a man standing down the street from the building calculates the angle of elevation to the top of the building to be 55o and the angle of elevation to the top of the flagpole to be 57o. Find the height of the building.
55o
tan 55oh
d
tan 55od h
45tan 57o
h
d
575.263 .ft
h
45
57o
d
tan 57 45od h
tan 57 45od h
tan 57 45 tan 55o od d tan 57 tan 55 45o od d tan 57 tan 55 45o od
45
tan 57 tan 55o od
45
tan 57 tan 55tan 55
o o
o h
Jeff Bivin -- LZHS
An observer standing on the cliff adjacent to the ocean looks out and sees an airplane flying directly over a ship. The observer calculates the angle of elevation to the plane to be 14o and the angle of depression to the ship to be 27o. How high above the ship is the airplane if we know that the ship is 1.5 miles from shore?
14o
b
tan(14 )1.5
o p
1.5 tan(14 )o p
1.5 tan 14 1.5 tan 27o o
1.138 miles
27o
p
tan(27 )1.5
o b
1.5 tan(27 )o b
Distance of plane above ship = p + b =
1.5 mi
Jeff Bivin -- LZHS
In Washington, D.C., the Washington Monument is situated between the Capitol and the Lincoln Memorial. A tourist standing at the Lincoln Memorial tilts her head at an angle of 7.491° in order to look up to the top of the Washington Monument. At the same time, another tourist standing at the Capitol steps tilts his head at a 5.463° to also look at the top of the Washington Monument. Find the distance from the Lincoln Memorial to the Washington Monument.
1.1tan(5.463 )o h
1.1tan(5.463 )o h
0.800 mi
tan(7.491 )o hd
tan(7.491 )od h
tan(7.491 )ohd
1.1tan(5.463 )
tan(7.491 )
o
od
