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Chapter 3 Sinusoids and Phasors

3.3 Inverse Trigonometric Functions

The notation cos – 1 y or arccos y is used to denote an angle whose cosine is . Thus, if y = cos , x
y
thenx = cos – 1 y . Similarly, if w = sin v , then v = sin – 1 w , and if z = tan u , then u = tan – 1 . z
These are called Inverse Trigonometric Functions.

Example 3.2

θ
Find the angle if cos – 1 0.5 = θ
Solution:

Here, we want to find the angle θ given that its cosine is 0.5. From (3.7), cos 60° = 0.5 . Therefore,
θ = 60°

3.4 Phasors

i
In the language of mathematics, the square root of minus one is denoted as , that is, i = – 1 . In
the electrical engineering field, we denote as to avoid confusion with current . Essentially, is
i
i
j
j
an operator that produces a 90° counterclockwise rotation to any vector to which it is applied as a
multiplying factor. Thus, if it is given that a vector has the direction along the right side of the
A
x -axis as shown in Figure 3.7, multiplication of this vector by the operator will result in a new
j
vector jA whose magnitude remains the same, but it has been rotated counterclockwise by 90° .
Also, another multiplication of the new vector jA by will produce another 90° counterclockwise
j
direction. In this case, the vector has rotated 180° and its new value now is A– . When this
A
)
vector is rotated by another 90° for a total of 270° , its value becomes j –( A = – jA . A fourth 90°
rotation returns the vector to its original position, and thus its value is again . Therefore, we
A
2 3 4
conclude that j = – 1 , j = j – , j = 1 , and the rotating vector is referred to as a phasor.
A
Note: In our subsequent discussion, we will designate the -axis (abscissa) as the real axis, and the
x
y -axis (ordinate) as the imaginary axis with the understanding that the “imaginary” axis is just as
“real” as the real axis. In other words, the imaginary axis is just as important as the real axis. *
An imaginary number is the product of a real number, say , by the operator . Thus, is a real
j
r
r
number and is an imaginary number.
jr
* A more appropriate nomenclature for the real and imaginary axes would be the axis of the cosines and the axis of the sines
respectively.

3−10 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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