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Dr. Hugh Blanton
ENTC 3331
Dr. Blanton - ENTC 3331 - Math Review
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Measurement Units
• The System of International Units (SI
units) was adopted in 1960.
• The use of older systems still persists,
but it is always possible to convert nonstandard measurements to SI units.
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SI (International Standard) Base Units
•
•
•
•
meter (m) = about a yard
kilogram (kg) = about 2.2 lbs
liter (l) = about a quart
liter (l) = 1000 mL
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Fundamental Units
Seven Fundamental physical phenomena.
Physical Property
Unit (abbreviation)
length
meter (m)
mass
kilogram (kg)
time
second (s)
electric current
ampere (A)
temperature
kelvin (K)
number of atoms or molecules mol (mol)
light Intensity
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candela (cd)
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Unit Conversions
•When converting physical values
between one system of units and
another, it is useful to think of the
conversion factor as a mathematical
equation.
• In solving such equations, one must
only multiply or divide both sides of the
equation by the same factor to keep the
equation consistent.
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Unit Conversions
•Example: 23 feet = ? yards
Same
 1 yd  23
 = 7.6
23 ft
yd
yd
Quantity
 3 ft  3
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Unit Conversions
•Example: 5 yd2 = ? ft2
2
2



9
3
ft
2
2
2  = 45 ft
5 yd 
1
yd


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Units
• Fundamental Units
• The SI system recognizes that there are
only a few truly fundamental physical
properties that need basic (and
arbitrary) units of measure, and that all
other units can be derived from them.
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• Derived Units
• The funadmental units are used as the
basis of numerous derived SI units.
• Note that derived SI units are
sometimes named after famous
physicists.
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Derived Units
Measureme
nt
Unit
Unit Description
force
newton (N) Force required to accelerate a mass of 1 kg at 1
m/s2
pressure
pascal
(Pa)
Pressure that exerts a force of 1 newton per m2 of
surface area
frequency
hertz (Hz)
Number of cycles of periodic activity per second
energy
joule (J)
Energy expended in moving a resistive force of 1
newton over 1 m.
power
watt (W)
Rate of energy expenditure of 1 joule per second.
electrical
charge
coulomb
(C)
Charge that passes a point in an electrical circuit if
1 ampere of current flows for 1 second.
electrical
resistance
ohm (W)
Ratio of the voltage divided by the current in an
electrical circuit.
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Unit Multiplication Factors
• An additional letter that denotes a
multiplying factor may prefix
fundamental or derived units.
• The more common multiplying factors
increase or decrease the unit by powers
of ten.
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Unit Multiplication Factors
• An additional letter that
denotes a multiplying
factor may prefix
fundamental or derived
units.
• The more common
multiplying factors
increase or decrease the
unit by powers of ten.
tera
(T)
1012
giga
(G)
109
mega
(M)
106
kilo
(k)
103
hecto
(h)
102
deca
(da)
10
deci
(d)
10-1
centi
(c)
10-2
milli
(m)
10-3
micro
(m)
10-6
nano
(n)
10-9
pico
(p)
10-12
10-15
femto
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Powers of Ten (big)
•101 = 10
•103 = 1000 (thousand)
•106 = 1,000,000 (million)
•109 = 1,000,000,000 (billion)
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Powers of Ten (small)
•100 = 1
•10-3 = 0.001 (thousandth)
•10-6 = 0.000001 (millionth)
•10-9 = 0.000000001 (billionth)
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Scientific Notation
• 7,000,000,000
• 7,000,000
• = 7 million
• = 7  106
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• = 7 billion
• = 7  109
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Scientific Notation
• 7,240,000
• = 7.24 million
• = 7.24  106
3 significant digits
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Very Large Quantities
• 7,240,000 = 7.24  106
6 decimal places
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Very Small Quantities
• 0.0000123 = 1.23  10-5
5 decimal places
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Engineering Notation
• Exponents = 3, 6, 9, 12, . . .
• Instead of 5.32  107
• we write
• 53.2  106
•
Decimal Exponent
part got
got
bigger
smaller
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Adding and Subtracting
•Exponents must be the same!
•(1.2  106) + (2.3  105)
•change to
•(1.2  106) + (0.23  106)
•= 1.43  106
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Multiplying
•Exponents Add
•(3.1  106)(2.0  102)
•= 6.2  108
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Dividing
•Exponents Subtract
•(3.8  106)
•(2.0  102)
•= 1.9  104
6-2=4
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Adding Fractions
•You can only add like to like
• Same Denominators
1 2 3
 =
5 5 5
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Different Denominators
•Make them the same
• find a common denominator
•The product of all denominators is
always a common denominator
• But not always the least common
denominator
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Finding the LCD
•Example:
1 4

12 15
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Factor the Denominators
12 = 2  2  3
15 = 3  5
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Assemble LCD
12 = 2  2  3
15 = 3  5
2  2  3  5 = 60
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Build up Denominators to LCD
1 ×5 4 ×4 5 16
 =

12 ×5 15×4 60 60
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Add Numerators
55 16
16 21
21 7
 == =
60
60 60
60 60
60 20
And Reduce if Needed
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Rational Expressions
•Example:
x -1
2x

2
2
x - 1 x - 2x  1
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Factor the Denominators
x - 1 = ( x  1)( x - 1)
2
x - 2x  1 =
( x - 1)( x - 1)
2
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DENOMINATORS
Assemble LCD
( x  1)( x - 1)
( x - 1)( x - 1)
( x  1)( x - 1)( x - 1)
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Build up Fractions to LCD
( x - )1(x -1)
2 x (x 1)

( x  1)( x - 1) ( x - 1)( x - 1)
(x -1)
(x 1)
FACTORED
LCD = ( x  1)( x - 1)( x - 1)
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Add Numerators
( x - 1)( x - 1)  2 x( x  1)
( x  1)( x - 1)( x - 1)
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Simplify Numerator
( x - 1)( x - 1)  2 x( x  1)
( x  1)( x - 1)( x - 1)
x - 2 x3x1 21x  2 x
( x  1)( x - 1)( x - 1)
2
2
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Radicals
Radical
Index
n
x
Radicand
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Meaning
n
x=y
if and only if
y =x
n
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Example
3
8=2
because
2 =8
3
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An Ambiguity
25 = 5
because
•but it’s also true
that. . .
5 = 25
2
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It’s also true that
( -5) = 25
2
•So why not say
•?
25 = -5
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Two Answers?
•Roots with an even index always have
both a positive and a negative root
•Because squaring either a negative or a
positive gives the same result
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Principal Root
•To avoid confusion we define the
principal root to be the positive root, so:
25 = 5 (not - 5)
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The Negative Root
•If we want the negative root we use a
minus sign:
- 25 = -5
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Negative Radicands
•Do Not Confuse
•With
-25
-25
- 25
•!!!
•Does not exist
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Negative Radicands
•You cannot take an even root of a
negative number
•Because you cannot square any number
and get a negative result
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Odd Roots of Negative Radicands
•You can take odd roots of negative
numbers:
-8 = -2 because
( -2)( -2)( -2) = -8
3
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Some Square Root Identities
2
x =x
•for all non-negative x
x =x
•for all non-negative x
x = x
•for all x
2
2
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A Common Error
a b  a  b
•for example, you cannot say
3  4 = 7 (WRONG!)
2
2
•What is the correct result?
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First Evaluate Inside
3 4
2
2
= 9  16
= 25
=5
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Products
•You can “split up” a radical when it
contains a product (not a sum!):
ab = a b
•(as long as a and b are non-negative)
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Example
400 = 16  25
= 16 25
= 4  5 = 20
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Perfect Squares
•Perfect squares are numbers that
have whole number square roots: 4, 9,
16, 25, 36, 49, 64, etc.
•All other numbers have irrational roots
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Numbers
• Natural Numbers: 1, 2, 3, . . .
• Whole Numbers: 0, 1, 2, 3, . . .
• Integers: . . . , -2, -1, 0, 1, 2, . . .
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Numbers
• Rational Numbers
• a/b (a,b integers, b not zero)
• Irrational Numbers
 Cannot be a ratio of integers
 Decimals never repeat or end.
 (decimals of rationals do)
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Rational and Irrational
3
Rational
= 0.75
(Terminates)
4
5
Rational
= 0.45454545
(Repeats)
11
2 = 1.41421356
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Irrational
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Numbers
• Real Numbers
 Rationals + Irrationals
 All points on number line
 All signed distances
 The Number Line
-6 -5 -4 -3 -2 -1 0
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2
3
4
5
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6
Imaginary Numbers
•Square root of a negative number
•We Define:
-1  i
Math, Physics
-1  j Engineering, Electronics
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Properties of j
j = -1
By Definition
j =-j
Because j 3= j 2j = (-1)j
j =1
Because j 4= j 2j 2 = (-1)(-1)
j = j
Because j 5= j 4j = (1)j
2
3
4
5
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Expressing Square Roots of Negative Numbers
-4 = (-1)4
-4 = -1 4
-4 = j 2 = 2 j
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Expressing Square Roots of Negative Numbers
-3 = (-1)3
-3 = -1 3
-3 = j 3
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Complex Numbers
•Real Part + Imaginary Part
•Example:
62j
Real Part = 6
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Complex Numbers
•Real Part + Imaginary Part
•Example:
62j
Imaginary Part = 2
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Adding and Subtracting Complex Numbers
•Likes stay with likes
• Re + Re = Re
• Im + Im = Im
•Just collecting like terms
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Adding and Subtracting Complex Numbers
•Example:
(6  2 j )  (2 - 3 j )
=8- j
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Adding and Subtracting Complex Numbers
•Example:
(6  2 j )  (2 - 3 j )
=8- j
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Adding and Subtracting Complex Numbers
•Example:
(6  2 j )  (2 - 3 j )
=8- j
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Multiplying
•Remember that j 2 = -1
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Multiplying
(6  2 j )(2 - 3 j )
= 12 - 18 j  4 j - 6 j
= 12 - 14 j - 6(-1)
= 18 - 14 j
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2
Dividing
•Complex Conjugate
• Reverse sign of imaginary part
Conjugate of
is
62j
6-2j
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Dividing
• Write as fraction
• Multiply numerator and denominator by
the complex conjugate of denominator
• Multiply and simplify
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Dividing
(6  2 j )  (2 - 3 j )
(6  2 j ) (2  3 j )
=
(2 - 3 j ) (2  3 j )
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Dividing
(6  2 j ) (2  3 j )
=
(2 - 3 j ) (2  3 j )
12  18 j  4 j  6 j
=
2
46 j -6 j -9 j
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2
Dividing
12  18 j  4 j  6 j
=
2
46 j -6 j -9 j
12  22 j - 6
=
49
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2
Dividing
12  22 j - 6
=
49
6  22 j 6 22
=
= 
j
13
13 13
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Graphing Complex Numbers
•Real part is x-coordinate
•Im. part is y-coordinate
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Graphing Complex Numbers
•Example: 3 + 2j  (3, 2)
Im
Re
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Polar Form
•Example: 3 + 2j  3.633.4°
Im
q
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Re
78
Polar Form
•Re + j Im  rq rejq
2
r = Re  Im
Re = r cosq
 Im 
q = tan  
 Re 
Im = r sinq
2
-1
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Trigonometric Form
•r (cos q + j sin q )
•Start with Re + j Im
•Substitute
•Re = r cos q
•Im = r sin q
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Trigonometric Form
•Start with Re + j Im
•Substitute
•r cos q + j r sin q
•r (cos q + j sin q )
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Euler’s Identity
e
re
jq
jq
= cosq  j sin q
= r (cosq  j sin q )
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Complex Arithmetic
•Addition & Subtraction
• Easiest in rectangular form
•Multiplication & Division
• Easiest in polar form
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Multiplication in Polar Form
•(r1q1) (r2q2)
•= r1r2 (q1+q2)
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Division in Polar Form
•(r1q1) / (r2q2)
•= r1 / r2 (q1-q2)
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Vectors & Scalers
• There is a fundamental distinction between
two types of quantity:
• Scalers and
• Vectors
• Scalers possess a magnitude, whereas vectors have
both magnitude and direction.
• Properties such as mass and temperature clearly have
no directionality and are examples of scalers.
• A complete description of force would be impossible
without specifying both the magnitude and direction of
the quantity.
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Vectors
•Represent magnitude and direction
•Example: Displacement
• “go 2 miles East”
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Vector Quantities
•Force
•Velocity
•Magnetic Field
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Vector Notation
•Vector: Bold or arrow over

F
•Scalar: Italic, no arrow
F
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Numerical Description
A vector can be represented in:
•Polar Form
• Magnitude and angle
•Rectangular Form
• x- and y-components
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Polar Form
V
Angle
V = (r, q )
V =(53, 65°)
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V = rq
V = 5365°
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Rectangular Form
V
Vy
q
Vx=V cos q
Vy=V sin q
Vx
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Rectangular to Polar
V
Vy
q
Vx
V = V V
2
x
2
y
V

y 
-1
q = tan  
V
 x
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Vector Addition
•Resultant vector
•Not the sum of the magnitudes
•Vectors add head-to-tail
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Vector Addition Example
•Go 3 miles East,
•then 4 Miles North
4
R = 5 miles at 53°
3
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Adding Nonperpendicular Vectors
•x-components add to give
x-component of resultant
•y-components add to give
y-component of resultant
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Adding Nonperpendicular Vectors
R=A+B
R
B
A
Rx = Ax + Bx
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Ry = Ay + By
97
Adding Nonperpendicular Vectors
Ry
R
By
Ay
B
A
Ax
Bx
Rx
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Trigonometric Functions
opposite
•Right Triangles Only!
q
adjacent
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adjacent
Trigonometric Functions
q
opposite
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Similar Triangles
q
q
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Same Angle
101
Similar Triangles
q
q
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Same Ratios
of Sides
102
Similar Triangles
•Ratios of sides depend ONLY on q
•So the ratio is a function of q
q
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Ratios of Sides
opp
sin q =
hyp
opposite
•Six Possible
adj
cos q =
hyp
q
adjacent
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opp
tan q =
adj
104
Ratios of Sides
opp
sin q =
hyp
hyp
csc q =
opp
adj
cos q =
hyp
hyp
sec q =
adj
opp
tan q =
adj
adj
cot q =
opp
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The Main 3 Trig Functions
SOHCAHTOA
opp
sin q =
hyp
adj
cos q =
hyp
opp
tan q =
adj
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Solving Triangles
•Find all 3 sides and 3 angles
•Need: 1 side plus 2 more items
• Only one more thing if it is given that
one angle is 90°
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Right Triangles
•Need 2 sides
•OR
•1 side and 1 angle
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Tool Kit
•The Trig functions
• (sin, cos, tan)
•The inverse Trig functions
• (sin-1, cos -1, tan -1)
•The Pythagorean Theorem
•Sum of angles is 180°
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The Trig Functions
•Find a side
•Given 1 side and 1 angle
opp
tan q =
adj
adj
cosq =
hyp
opp
sin q =
hyp
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The Inverse Trig Functions
•Find an angle
•Given 2 sides
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The Pythagorean Theorem
•Find a side
•Given 2 sides
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Angles add to 180°
•Find an angle
•Given the other angle
90 - q
q
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Vector Multiplicaton
• Three types vector multiplication:
• Simple multiplication
• Dot Product
• Always yields a scaler answer.
• Cross Product
• Always gives a vector result.
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Dot Product
 
A  B = AB cos AB
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z
x
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
A = 2 2  32  32 = 22
z

A xˆ 2  yˆ 3  zˆ3
 =
22
A

A
x
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