MATLAB Trigonometry, Complex Numbers and Array Operations

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MATLABTrigonometry, Complex Numbers and Array
Operations
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Basic Trigonometry
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Basic Trigonometric Expressions in MATLAB

• sin(alpha) Sine of alpha
• cos(alpha) Cosine of alpha
• tan(alpha) Tangent of alpha
• asin(z) Arcsine or inverse sine of z, where z
  must be between .1 and 1. Returns an angle
  between .p/2 and p/2 (quadrants I and IV).
• acos(z) Arccosine or inverse cosine of z, where z
  must be between .1 and 1. Returns an angle
  between 0 and p (quadrants I and II).
• atan(z) Arctangent or inverse tangent of z.
  Returns an angle between .p/2 and p/2 (quadrants
  I and IV).
• atan2(y,x) Four quadrant arctangent or inverse
  tangent, where x and y are the coordinates in the
  plane shown in the .gure above. Returns an angle
  between -p and p (all quadrants), depending on
  the signs of x and y.

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Hyperbolic Functions

• The hyperbolic functions are functions of the
  natural exponential function ex, where e is the
  base of the natural logarithms, which is
  approximately e 2.71828182845904. The inverse
  hyperbolic functions are functions of the natural
  logarithm function, ln x.
• The curve y cosh x is called a catenary (from
  the Latin word meaning chain). A chain or rope,
  suspended from its ends, forms a curve that is
  part of a catenary.

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Basic Hyperbolic Expressions in MATLAB
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Complex Numbers

• Imaginary number The most fundamental new
  concept in the study of complex numbers is the
  imaginary number j. This imaginary number is
  defined to be the square root of -1
• j v(-1)
• j2 -1

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Complex Numbers - Rectangular Representatiton
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Complex Numbers - Rectangular Representatiton

• Rectangular Representation A complex number z
  consists of the real part x and the imaginary
  part y and is expressed as
• z x jy
• where
• x Rez y Imz

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Complex Numbers - Rectangular Representatiton

• A general complex number can be formed in three
  ways
• gtgt z 1 j2
• z
• 1.0000 2.0000i
• gtgt z 1 2j
• z
• 1.0000 2.0000i
• gtgt z complex(1,2)
• z
• 1.0000 2.0000i

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Complex Numbers - Rectangular Representatiton

• gtgt z 3 4j
• z
• 3.0000 4.0000i
• gtgt x real(z)
• x
• 3
• gtgt y imag(z)
• y
• 4

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Polar Representation

• Defining the radius r and the angle ? of the
  complex number z can be represented in polar form
  and written as
• z r cos ? jr sin ?
• or, in shortened notation
• z r ?

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Polar Representation

• gtgt z 3 4j
• gtgt r abs(z)
• r
• 5
• gtgt theta angle(z)
• theta
• 0.9273
• theta (180/pi)angle(z)
• theta
• 53.1301

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Polar Representation
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Arrays and Array Operations

• Scalars Variables that represent single numbers,
  as considered to this point. Note that complex
  numbers are scalars, even though they have two
  components, the real part and the imaginary part.
• Arrays Variables that represent more than one
  number. Each number is called an element of the
  array. Rather than than performing the same
  operation on one number at a time, array
  operations allow operating on multiple numbers at
  once.
• Row and Column Arrays A row of numbers (called a
  row vector) or a column of numbers (called a
  column vector).
• Two-Dimensional Arrays A two-dimensional table
  of numbers, called a matrix.
• Array Indexing or Addressing Indicates the
  location of an element in the array.

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Vector Arrays

• Consider computing y sin(x) for 0 x p. It
  is impossible to compute y values for all values
  of x, since there are an infinite number of
  values, so we will choose a finite number of x
  values.
• Consider computing
• y sin(x), where x 0, 0.1p, 0.2p, . . . , p
• You can form a list, or array of the values of
  x, and then using a calculator you can compute
  the corresponding values of y, forming a second
  array y.

x and y are ordered lists of numbers, i.e., the
first value or element of y is associated with
the first value or element of x. Elements can be
denoted by subscripts, e.g. x1 is the first
element in x, y5 is the fifth element in y. The
subscript is the index, address, or location of
the element in the array.
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Vector Creation

• By an explicit list,
• gtgt x0 .1pi .2pi .3pi .4pi .5pi .6pi .7pi
  .8pi .9pi pi
• By a function,
• gtgtysin(x)
• Vector Addressing
• A vector element is addressed in Matlab with an
  integer index (also called a subscript) enclosed
  in parentheses. For example, to access the third
  element of x and the ?fth element of y
• gtgt x(3)
• ans
• 0.6283
• gtgt y(5)
• ans
• 0.9511

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Colon Notation

• Addresses a block of elements The format for
  colon notation is
• (startincrementend)
• where start is the starting index, increment is
  the amount to add to each successive index, and
  end is the ending index, where start, increment,
  and end must be integers. The increment can be
  negative, but negative indices are not allowed to
  be generated. If the increment is to be 1, a
  shortened form of the notation may be used
• (startend)

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Colon Notation Examples

• 15 means start with 1 and count up to 5.
• gtgt x(15)
• ans
• 0 0.3142 0.6283 0.9425 1.2566
• 7end means start with 7 and count up to the
  end of the vector.
• gtgt x(7end)
• ans
• 1.8850 2.1991 2.5133 2.8274 3.1416
• 3-11 means start with 3 and count down to 1.
• gtgt y(3-11)
• ans
• 0.5878 0.3090 0
• 227 means start with 2, count up by 2, and
  stop at 7.
• gtgt x(227)
• ans
• 0.3142 0.9425 1.5708