Download Trigonometry Part I

Math 416
Trigonometry
Time Frame
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1)
2)
3)
4)
5)
6)
7)
8)
Pythagoras
Triangle Structure
Trig Ratios
Trig Calculators
Trig Calculations
Finding the angle
Triangle Constructions
Word Problems
Right Angle Triangles
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The next section will deal exclusively
with right angle triangles. We recall
Pythagoras
x2 = y2 + z2
β
x
y
z
Angle Sum
θ
θ+ β + = 180°
Pythagoras
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Example
72 = x2 + 32
7
x
49 = x2 + 9
40 = x2
6.32 = x
3
x = 6.32
Do Stencil #1 
Triangle Structure
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We all need to agree on what we are
talking about. Consider
A
<BAC = <A
<ABC = <B
b
c
<BCA = <C
B
a
Do Stencil #2 
C
AB = c
opposite
BC = a
opposite
CA = b
opposite
Trig Ratios
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When we consider the similarity of right
angle triangles as long as we ignore
decimal angles there are only 45 right
angle triangles.
Consider the angles
90° – 1° – 89°
90° – 2°– 88°
90° – 3° – 87° …
90° – 45° - 45°
Then we start over
Trig Ratios
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From ancient times, people have
looked at the ratios within right
angles triangles
First in tables
Now stored in calculators
We need to define the parts of a
right angle triangle
Two types of definitions
Definitions
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Absolute – never changes
Relative – involves the position
Absolute vs Relative
A
β
Now we can define absolutely
the hypotenuse as the side
opposite the right angle (longest
side). In this example it is side
AC or b.
b
c
B
Now “relative to angle θ”
we define side AB or c as
the opposite side
a
θ
Now “relative to angle β”
we define side BC or a as
C the opposite side
Absolute Vs. Relative
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Now “relative to angle θ” we define
side BC or a as the adjacent side
Now “relative to angle β” we define
side AB or c as the adjacent side
Labeling the Triangle
Hence with respect to θ
Hyp
Opp
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θ
Adj
Now we
define the
three main
trig ratios…
Trig Ratios
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The sine of an angle is defined as the
ratio of the opposite to the
hypotenuse. Thus Sin θ= Opp
Hyp
The cosine of an angle is defined as
the ratio of the adjacent to the
hypotenuse. Thus Cos θ= Adj
Hyp
Trig Ratios
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The tangent of an angle is defined as
the ratio of the opposite to the
adjacent. Thus Tan θ = Opp
Adj
SOH – CAH - TOA
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You may of heard the acronym SOH
– CAH – TOA or SOCK – A – TOA
Sin Opp
Hyp
Cos Adj
Hyp
Tan Opp
Adj
Old Harry And His Old Aunt
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There is another acronym… old Harry and
his old aunt
Sin Opp
Hyp
Cos Adj
Hyp
Tan Opp
Adj
Use the acronym that you can remember
Example
Consider
A
39
39
36
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Sin A = 15 Sin C = 36
B
15
39
Cos A = 36 Cos C = 15
39
39
C
Tan A = 15 Tan C = 36
36
15
Trig Calculator
Now note the table for the
assignment is as follows (question
B #3). For example
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40
24
#
Eg
A
32
Angle
B
C
Sin Cos Tan Angle Sin Cos Tan
32 24 32 C
24 32 24
40 40 24
40 40 32
Trig Calculator
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We note that these ratios are stored
by angle albeit as decimals in a
calculator
Note first and foremost your
calculators
IT MUST BE IN DEGREES
Make sure you find your DRG
(Degree – Radian – Gradients)
Trig Calculator
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Hence if θ = 54° then to 4 decimal
places
Sin 54° = 0.8090
•Cos 54° = 0.5878
•Tan 54° =
1.3764
Do Stencil #3 
Question #4
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The table required for #4 is as
follows
Example θ = 37°
#
Sin
Cos
Tan
Eg
0.6018 0.7986 0.7536
Trig Calculations
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There are three basic type of
questions. We will focus on the Sine
ratio (like question #5) but the
techniques are the same for all trig
ratio problems.
Trig Calculations
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Solve for x
Consider
Use the angle given to you!
12
x
40°
Step #1: Determine
the Trig Ratio
involved with
respect to the angle
12 = hypotenuse, x = opp Thus, SINE
Trig Calculations
Step #2 – Determine the
equation
12
X = sin 40°
12
x
40°
Step #3: Cross multiply (if necessary)
x = 12 Sin 40°
Trig Calculations
Step #4 If the unknown is
isolated (by itself) solve… if not
divide then solve
12
x
40°
x = 7.71
Trig Calculations
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More Practice
x = sin 39°
11
x
11
x = 11 sin 39°
39°
x = 6.92
Trig Calculations
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Even More Practice
11 = sin 42°
x
x
11
11 = x sin 42°
42°
x = 16.44
Divided both sides by sin 42° or 0.67
Trig Calculations
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Even More Practice
9 = sin 73°
x
x
9
73°
x=
9
.
sin 73°
x = 9.41
Finding the Angle
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Up until now we have the angle get the
ratio
Now we need to go the other way
Given the ratio, give the angle
Eg. The buttons we are looking for are the
inverse sine (sin -1)
Inverse cosine (cos -1)
Inverse tangent (tan -1)
Find it on your calculator
Examples of Finding the Angle
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Find the angle
Sin θ = 5
16
16
θ= sin
-1
5
θ
(5)
16
θ= 18°
(no decimals)
Another Example
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Find the angle
sin θ = 7
31
31
7
θ
θ= 13°
Other Examples
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Now all the Trig Calculations can
follow these procedures
x = cos 25°
15
15
x = 13.59
25°
x
Sin, Cos or Tan?
Another Example
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Find the Side
x = Tan 61°
6
x = 6 tan 61°
x
61°
6
x = 10.82
Another Example
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Find the angle
sin θ = 7
31
31
sin θ ( 7 )
7
θ
31
θ= 13°
Another Example
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Find the angle
cos θ = 5
7
7
θ = 44°
θ
5
Another Example
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Find the angle
tan θ = 18
5
θ = 74°
18
θ
5
Completing the Triangle
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Now using our knowledge we can
complete triangles
76°
x
y
Draw this triangle and
another one right below…
fill out missing info
5 = cos 14° y = tan 14°
14° x
5
x = 5.15
5
y = 1.25