Download 1 Lecture 16 Right Triangle Trigonometry All Right Triangles with a

Lecture 16
Right Triangle Trigonometry
55º
8 feet
5 ft
How high is Sparty?
1
© m j winter, 2000
All Right Triangles with a 55º Angle Are Similar
C”
C’
C
A
55o
55o
55o
A
B
CB/AB
=
C’B’/AB’
B”
A
B’
=
C”B”/AB” = tan55 o
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1
Basic Trigonometric Functions
(Tri -Gon = 3 sides)
b
opposite
=
c hypotenuse
a
adjacent
cos(θ ) = =
c hypotenuse
sin(θ ) =
tan(θ ) =
b opposite
=
a adjacent
b = opposite
c = hypotenuse
q
a = adjacent
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All Right Triangles with a 55º Angle Are Similar
tan(55º) = 1.42814...
55º
8
feet
y
8
55 o
5 ft
5
y
tan(55o )=1.42814
8=
y = 8(1.42814) =11.425
Sparty = y+ 5 = 16.425
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2
Values of Trig. Functions
Two cases when exact answers are used:
30o-60o-90o triangle and 45 o-45o-90o triangle
s
30
s
s
h
60
o
sin60
h
2
h2 = s 2 −
h=s
o
s/2
s 
h2 +   = s 2
2
2
2
s
3s
=
4
4
3
2
3
2 = 3
s
2
s
1
cos60 = 2 =
s 2
60
s
=
s
3
2
tan60 =
= 3
s
2
1
sin30o = cos60o =
2
o
Notice that s
does not appear
in the values of
the functions
s
cos30o = sin60o =
3
2
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45o-45o-90o triangle
2
2
s +s =c
2
sin45o = cos45o =
2 s 2 = c2
s
1
2
=
=
2
s 2
2
s 2 =c
45
For all other angles, use a calculator.
c
s
Make sure it is in degree mode!
45
s
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3
How to Remember These
θ
Sin( θ)
0
o
30
o
45
o
60
o
90
Cos(θ)
0
2
1 1
=
2 2
2
2
3
2
4
=1
2
4
=1
2
3
2
2
2
1 1
=
2 2
0
2
Tan(θ)
0
1
3
1
3
Not defined
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Applications - 1
Angle of elevation Angle of Depression
If you are sitting 20 feet up in a tree stand and see a deer at an
angle of depression of 42 degrees, how far is the deer from the
base of the tree? How far is the deer from you? Assume the tree
is vertical.
you
42 o
20’
deer
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4
Solution to Deer problem
you
x
= tan48o
20
42 o
48
y
20’
x = 20 ⋅ tan48o = 22.2122feet
deer
x
Can find y by using trigonometry or by using Pythagorean theorem
x 2 + 202 = y2
or 20/y = cos(48o)
y = 29.89 feet
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Piccadilly Circus
The traffic at Piccadilly Circus is so bad that you cannot get very close to
the statue in the middle. From a spot on a side street, you look at the
statue and note that the angle of elevation is about 45º.
You walk about 20 feet closer and observe that the angle has now
become 60º.
How high is the statue?
10
5
PC - diagrams
Firstly, make a diagram and label
all known quantities
Second: label unknown parts
h
45º
60
45º
20
20
60
x
11
45º
20
60
x
PC - analysis - Two triangles
h h
= tan60o
x
h
= tan45o
20 + x
h
45º
20
We want to find h, so we’ll solve both
equations for x and equate them.
60
x
h
45º
20
60
12
x
6
h
= tan60o
x
h
= tan45o
20 + x
PC - analysis - Two triangles
h
h
x=
=
3
tan60o
h
20 + x =
tan45o
x = h − 20
=
h
1
Solve for h:
h
= h − 20
3
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PC - finish
h
= h − 20
3
 3 − 1

1 
20 = h  1−
=
h



3
3 


h=
20 3
= 47.32
3 −1
Statue is 47.32 feet higher than your head.
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7
Inverse Trig Functions
How do we solve these:
cos(x) = .456
sin(x) = .321
tan(x) = 1.5
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Inverse Trig Functions
To solve:
cos(x) = .456
sin(x) = .321
tan(x) = 1.5
Use the inverse trig functions!
Old fashioned language
arccos(x) means “the angle whose
cosine is x”
arcsin(x) means “the angle whose
sine is x”
arctan(x) means “the angle whose
tangent is x”
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8
Inverse Trig Functions - 2
To solve:
cos(x) = .456
sin(x) = .321
tan(x) = 1.5
Modern usage:
arccos(x) = “the angle whose cosine is x” becomes
cos–1(x), the inverse cosine of x.
arcsin(x) = “the angle whose sine is x” becomes
sin–1(x), the inverse sine of x.
arctan(x) = “the angle whose tangent is x” becomes
tan–1(x), the inverse tangent of x.
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Inverse Trig Functions - 3
To solve:
cos(x) = .456
sin(x) = .621
tan(x) = 1.5
Modern usage:
arccos(x) = “the angle whose cosine is x” becomes
cos–1(x), the inverse cosine of x.
arcsin(x) = “the angle whose sine is x” becomes
sin–1(x), the inverse sine of x.
arctan(x) = “the angle whose tangent is x” becomes
tan–1(x), the inverse tangent of x.
cos–1(.456) = 62.87 o
sin–1(.621) = 38.39o
tan–1(1.5) = 56.31o
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9
Pythagoras and Trigonometry
c
a
θ
b
a2 + b 2 = c2
a2 + b 2 = 1
c2 c2
2
2
a
b
=1
c + c
(sin(θ)) 2 + (cos(θ))2 = 1
sin 2(θ) + cos 2(θ) = 1
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