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(x + 2)(x2 + 3)(2 x 7) ExpandNumerator ExpandDenominator 2 (x + 5 x + 2)(x 5)(x + 6) 42 9 x 8x2 3x3 + 2x4 60 148x 23x2 + 6x3 + x4
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(x + 2)(x2 + 3)(2 x 7) / /Together ExpandAll 2 (x + 5 x + 2)(x 5)(x + 6) 42 9x 8x2 3x3 + 2x4 60 148x 23x2 + 6x3 + x4
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x2 2x + 3 7x 2 , , and 2 and express as a single fraction with expanded numerator and 5x 7 3x + 1 x +1 denominator.
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5 x 7 7x 2
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x2 ; x2 + 1
Algebra and Trigonometry
Together[p + q + r] //ExpandDenominator
17 48x + 51x2 64x3 + 56x4 7 16x + 8x2 16x3 + 15x4
Without //ExpandDenominator, the denominator would be expressed in factored form.
7.16 What is the partial fraction expansion of
SOLUTION
6 (x 1) Apart 2 2 (x +1)(x +1)(x 4)
( x 1)6 ( x + 1)( x + 1)2 ( x 4)
4 +
4(4 + x) 729 32 288 +x 2 + 425( 4 + x) 5(1 + x) 25(1 + x) 17(1 + x2)
7.17 Find the partial fraction expansion of the function in the previous problem with linear complex denominators.
SOLUTION
6 (x 1) Apart 2 (x + I)(x I)(x +1)(x 4)
To force Mathematica to express the result using linear complex denominators, we factor x2 + 1 as (x + I ) (x I ).
2 - 8 2 + 8 32 729 288 + x 17 17 17 17 2 + +x 425( 4+ x) +x 5(1+ x) 25(1+ x)
7.18 Express (ex + e2x)4 as a sum of exponentials.
SOLUTION
Expand[(Ex + E2 x)4] 4 x + 4 5 x + 6 6 x + 4 7 x + 8 x
7.3 Trigonometric Functions
Although the commands discussed in the previous section may be applied to trigonometric functions, doing so does not take advantage of the simplification offered by trigonometric identities. To incorporate these into the calculation, the option Trig True must be set. (The default is Trig False for all but the Simplify command.) The following examples show the difference.
EXAMPLE 21
Cancel Sin[x] 2 1 Cos[x] Sin[x] 1 Cos[x] x2 Cancel Sin[x] 2 , Trig True 1 Cos[x] Csc[x] s
EXAMPLE 22
2 2 Together Cos[x] 2 + Sin[x] 2 1 Sin[x] 1 Cos[x] 2 4 2 4 Cos[x] Cos[x] + Sin[x] Sin[x] 2 2 ( 1 + Cos[x] ) ( 1 + Sin[x] )
Algebra and Trigonometry
2 2 Together Cos[x] 2 + Sin[x] 2 , Trig True 1 Sin[x] 1 Cos[x]
2 Trig True applies to hyperbolic as well as circular functions.
EXAMPLE 23
Expand[(Cosh[x]2 + Sinh[x]2)(Cosh[x]2 Sinh[x]2)] Cosh[x]4 Sinh[x]4 Expand[(Cosh[x]2 + Sinh[x]2)(Cosh[x]2 Sinh[x]2), Trig True] Cosh[x]2 + Sinh[x]2
To allow additional manipulation of trigonometric expressions, Mathematica offers the following specialized commands, which apply to both circular and hyperbolic functions:
TrigExpand[expression] expands expression, splitting up sums and multiples that appear in arguments of trigonometric functions and expanding out products of trigonometric functions into sums and powers, taking advantage of trigonometric identities whenever possible. TrigReduce[expression] rewrites products and powers of trig functions in expression as trigonometric expressions with combined arguments, reducing expression to a linear trig function (i.e., without powers or products). TrigFactor[expression] converts expression into a factored expression of trigonometric functions of a single argument.
The next example shows the difference between Expand and TrigExpand.
EXAMPLE 24
Expand[(Sin[x] + Cos[x])2] Cos[x]2 + 2 Cos[x] Sin[x] + Sin[x]2 TrigExpand[(Sin[x] + Cos[x])2] 1 + 2 Cos[x] Sin[x]
EXAMPLE 25
TrigExpand[Sin[x + y]] Cos[y] Sin[x] + Cos[x] Sin[y] TrigExpand[Sin[2 x]] 2 Cos[x] Sin[x] TrigExpand[Sin[2 x + y]] 2 Cos[x] Cos[y] Sin[x] + Cos[x]2 Sin[y] Sin[x]2 Sin[y]
TrigExpand can also be applied to hyperbolic functions.
EXAMPLE 26
TrigExpand[Cosh[x + y]] Cosh[x] Cosh[y] + Sinh[x] Sinh[y]
EXAMPLE 27
TrigReduce[Sin[2 x] + Sin[x] Cos[3 x] ] 1 (4 4 Cos[4x] 3 Sin[2x]+ 3 Sin[4x] Sin[8x]+ Sin[10x]) 8
TrigReduce rewrites the original expression as a linear trig expression.
Algebra and Trigonometry
TrigReduce[Sinh[2 x]2 + Sinh[x] Cosh[3 x]3] 1 ( 4 + 4 Cosh[4x] 3 Sinh[2x]+ 3 Sinh[4x] Sinh[8x]+ Sinh[10x]) x 8
The next example shows the difference between TrigFactor and TrigReduce. Notice that TrigFactor writes the expression as a product, while TrigReduce writes the expression as a sum of linear trig functions.
EXAMPLE 28
expression = 24 Sin[x]2 Cos[x]2 + 16 Cos[x]4; TrigFactor[expression] 4 Cos[x]2 ( 5 + Cos[2 x]) TrigReduce[expression] 9 + 8 Cos[2 x] Cos[4 x]
The Solve command can be used to solve trigonometric equations. However, because only principal values of inverse trigonometric functions are returned, not all solutions will be obtained.
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