Squares and Square Roots
A square is a number multiplied by itself. For example, 2
x
2 = 4.
Therefore, 4 is the square of 2. Likewise, 5
x 5 = 25 and 10
x 10 =
100. Therefore, the square of 5 is 25 and the square of 10 is 100.
A square can also be notated as a number raised to the power of 2
(raised to an exponent of 2). This is notated as 3
^{2} .
Therefore, the square of 3 can be expressed as follows:
3 squared = 3 x 3 = 3^{2}
= 9
A square root is the opposite of a square. If 9 is the
square of 3 then 3 is the square root of 9. Also, as in the examples
above the square root of 25 is 5 and the square root of 100 is 10. The
symbol for square root is √ .Therefore, the following
expresses that the square root of 25 is 5:
One practical use of squares and square roots is with the area of a
square. If you take the mathematical square of the length of one side
of a physical square you get the area of the square. The square root of
the area of a physical square is the
length of one side of the square. A square that is 10 inches
on each side has an area of 100 square inches. Likewise, a square with
an area of 100 square inches has four sides that are 10 inches long
each.
Geometry of Right Triangles
A right angle is formed by two lines that meet at an angle of 90
degrees; a corner of a square forms a
right angle. A right triangle is a triangle that has one angle that is
90 degrees, like a corner cut from a square.
Right angle geometry comes into play when working with AC circuits.
When inductors, capacitors and resistors work together in circuits, the
timing of the voltage and current across the components (phase angle)
are such that right triangle geometry can be used to analyze
the relationships.
The primary formula for working with right triangles is known as the
Pythagorean Theorem. This theorem first names the three lines that make
up the right triangle. One of the two lines that form the right angle
is called the base. The other line that forms the right angle is called
the adjacent. The line that joins these two lines is called the
hypotenuse.
The lines of a right
triangle 
The Pythagorean Theorem is stated as follows:
The square of the hypotenuse is equal
to the sum of the squares of the other two sides.
To find the length of the hypotenuse:
 Find the square of the length of the base
 Find the square of the length of the adjacent
 Add these squares together
 Find the square root of the sum
For example:
In this example:
 5 squared (5^{2} or 5 X 5) = 25
 2 squared (2^{2} or 2 X 2) = 4
 25 + 4 = 29
 The square root of 29 (√29) = 5.39
To calculate this on a scientific calculator, use the following
keystrokes (The last number, without the gray background is the answer.
You don't type that in.) :
5


X^{2}


+


2


X^{2}


=


√ 

5.39


press these keys

displayed
answer

Complex Numbers
A complex number is two numbers that are added together after
multiplying one of the numbers by the square root of 1. This isn’t as
daunting as it might sound. First of all, the square root of 1 is
represented by the lowercase letter i. So a complex number looks like
the following:
A + iB
A complex
number
Complex numbers are useful in electronics because they can be used to
describe right triangles. The right triangle above can be notated as:
5 + i2
The complex number simply says "Go 5 units, turn right then
go 2 units". If the triangle is flipped, as follows:
This triangle can be notated as:
5  i2
This can be interpreted as "Go 5 units, turn left then go 2 units". +i
means turn right and i means turn left.
In electronics, to avoid confusing complex numbers with the symbol
for current the lower case j is used in place of the lower case I. If
the above triangles were being used to
analyze an AC circuit, they would be notated as follows:
Complex numbers using j instead of i.
5 + j2 

5  j2 
First
Triangle 

Second
Triangle 
When AC circuits are expressed with complex numbers it is called
rectangular notation. In this case, the base of the triangle represents
resistance and the adjacent represents capacitive reactance or
inductive reactance. Capacitive reactance is represented by  j
and inductive reactance is represented by + j. Since inductive
reactance and capacitive reactance cancel each other and +j and j
also cancel each other — for example (+ j10) + ( j10) = 0 — complex
numbers
facilitate adding capacitive reactance and inductive reactance.
Sine, cosine and tangent
Describing two sides of a right triangle with rectangular
notation is sufficient to describe the whole triangle. Another way to
describe a right triangle is to describe the length of the hypotenuse
and the angle between the hypotenuse and the base. This is called polar
notation and is also
useful when working with AC circuits. The triangle above would be
described in polar notation as follows:
The triangle from above
described with polar
notation. 
To convert from polar notation to rectangular notation, you need to
find the sine and the cosine of the angle between the base and the
hypotenuse.
Sine
The sine of an angle is the length of the adjacent divided by the
length of the hypotenuse as shown in the following formula (
θ, the Greek letter Theta represents
an angle):
In our current example the adjacent is 2 inches and the hypotenuse is
5.39 inches. This gives us 2 ÷ 5.39 which is
0.371. Therefore, the sine of an angle of 21.8 degrees is 0.371. This
is true no matter what the actual lengths of the sides are.
The adjacent is the length of the
hypotenuse multiplied by the sine of the angle.
a = h x sin θ
Before pocket calculators, mathematicians would find the sine of an
angle by looking it up in a book of tables. Today, all you have to do
is press the [sin] button on a calculator. To calculate the
length of the adjacent of the preceding triangle, use the following
sequence:
2


1


.


8


sin


X


5


.


3


9


=


2

This will give the answer of 2 inches.
Cosine
The cosine of an angle is the length of the base divided by the length
of the hypotenuse. In this example that is 5 ÷ 5.39 which is 0.928.
Therefore, the cosine of an angle of 21.8 degrees is 0.928.
Multiplying the cosine of the angle by the length of the hypotenuse
gives you the length of the base.
b = cos θ x h
Therefore, to calculate the length of the base of the preceding
triangle, use the following sequence:
2


1


.


8


cos


X


5


.


3


9


=


5

This will give the answer of 5 inches.
In review, if you have the length of the hypotenuse of a triangle and
the angle between the hypotenuse and the base, you can calculate the
lengths of the adjacent and the base with two simple formulas. The
length of the base is the length of the hypotenuse multiplied by the
cosine of the angle and the length of the adjacent is the length of the
hypotenuse multiplied by the sine of the angle.
Relationships of the sides of a
right triangle.

Tangent
The tangent of an angle is the length of the adjacent divided by the
length of the base. In the example, that is 2 ÷ 5 which is 0.4.
Therefore, the tangent of an angle of 21.8 degrees is 0.4.
Arcsine, arccosine and arctangent
These are simply the reverse (actually the inverse) of the sine, cosine
and tangent. The arcsine of 0.371 is 21.8 degrees,
the arccosine of 0.928 is 21.8 and the arctangent of 0.4 is 21.8
degrees.
Most calculators label the arcsine button with sin
^{1},
arccosine with
cos
^{1} and arctangent with tan
^{1}. Often you must
press an "inverse"
button before using these buttons. For example, to find the arcsine you
may have to press [inv] then [sin]. Once again, be sure to familiarize
yourself with your own calculator.
Conversion between polar and rectangular notation
You now have the tools to convert between polar and rectangular
notation.
To convert from rectangular notation to polar notation, first use the
Pythagorean theorem to find the length of the hypotenuse. Then, divide
the adjacent by the base and find the arctangent of the result.
Therefore, in the current example, the rectangular notation is 5 + j2,
where 5 is
the base and 2 is the adjacent. To convert this to polar notation, use
the following sequence.
5


X^{2}


+


2


X^{2}


=


√ 

5.39

This gives you the length of the hypotenuse which is 5.39 inches.
Next, divide the adjacent by the base then find the arctangent of the
result:
This gives you the angle between the base and the hypotenuse, which is
21.8 degrees.
To convert from polar notation to rectangular notation:
 Find the
length of the base by multiplying the length of the hypotenuse by the
cosine of the angle
 Find the length of the adjacent by the
multiplying the length of the hypotenuse by the sine of the
angle
On the calculator this is...
2


1


.


8


sin


X


5


.


3


9


=


5

and
2


1


.


8


cos


X


5


.


3


9


=


2

Which gives you 5 and 2 respectively which in polar notation is 5 + j2
Vectors
A vector represents a value using an arrow with a
length that is proportional to that value. This arrow points in a
particular direction showing that the value has something to do with
direction. By definition, a vector is quantity that
has both magnitude and direction. For example I can represent walking
100 feet to the north by drawing an arrow that is 100 mm long and
points to the top of the paper. Two
vectors can be combined to get a third vector that represents the
result of the first two values. For example, let's say that after
walking 100 feet to the north I turn right and walk 100 feet to the
east. I can represent this action by drawing another arrow from the
point of the first that goes 100 mm to the right. The result is as if I
had walked 141 feet to the northeast. The third vector would be an
arrow that starts at the same place as the first and goes 141 mm at a
45 degree angle.
If you walk 100 feet north then
100 feet east you end up in the same place as if you walked 141 feet
northeast. "The above vectors represent the trip using millimeters to
represent
feet.

For another example, assume an airplane is flying on a heading of 0
degrees (northward) at 200 mph. The
magnitude of the vector is 200 mph and the direction is 0 degrees. If
the wind is
blowing at 50 mph to the east (90 degrees) the airplane will be blown
offcourse. You can see how far the plane is blown offcourse by
drawing vectors on
a grid. For example, you can draw an arrow along one line that is 20
squares long, representing 200 mph to represent the speed and direction
of the airplane. At
the point of this arrow you can draw another arrow pointing to the
right with a length of 5 squares representing 50 mph. The point of this
arrow will
represent the position of the airplane after flying under these
conditions for one hour. The airplane has will go 200 miles to
the north and 50 miles to the east, ending up
50 miles east of where the pilot expected.
Using vectors to plot the course of an airplane 
If you draw a line between the starting point and the ending point you
will get the actual course the airplane traveled. Using a ruler and
protractor you can measure the length of this line and the angle from
the intended course. In this example the airplane traveled over the
ground at a speed of 206 mph on a course of 14 degrees east.
As shown above, you can calculate the distance the airplane actually
traveled using the Pythagorean theorem. The actual course can be
calculated with trigonometry. In this case, to find the angle of the
hypotenuse of the triangle (the actual course), divide the wind speed
(the adjacent of the triangle) by the airplane speed (the base of the
triangle) then calculate the arctangent of the result. This gives
an angle of approximately 14 degrees.
course
= arctan

wind
speed 
————————————— 
airplane speed

On the calculator that would be:
5


0


÷ 

2


0


0


=


tan^{1}


14

Which gives an angle of 14 degrees.
Using vector analysis is just a rework of the trigonometry that was
already covered. The interaction of resistance, capacitive reactance
and inductive reactance in an AC circuit can be analyzed using vectors.
There are other aspects of electronics that can be analyzed with
vectors too, such the interactions of different frequencies.