# sql server reporting services barcode font Consider the equation 1 2 cos x sin x + sin 2x = 0. in Software Generating QR Code in Software Consider the equation 1 2 cos x sin x + sin 2x = 0.

EXAMPLE 29 Consider the equation 1 2 cos x sin x + sin 2x = 0.
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equation = 1 2 Cos[x] Sin[x] + Sin[2 x] 0 Solve[equation , x]
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Solve ifun : Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
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Since trigonometric and hyperbolic functions can be represented in terms of exponential functions (complex exponentials in the case of circular trig functions), Mathematica offers two conversion functions: TrigToExp[expression] converts trigonometric and hyperbolic functions to exponential form. ExpToTrig[expression] converts exponential functions to trigonometric and/or hyperbolic functions.
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TrigToExp and ExpToTrig may also be used to convert inverse trigonometric and hyperbolic functions.
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EXAMPLE 30
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TrigToExp[Cos[x]] + 2 2 TrigToExp[Sinh[x]] + 2 2 ExpToTrig[Exp[x]]
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Cosh[x] + Sinh[x] ExpToTrig[Exp[I x]] Cos[x] + Sin[x]
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Algebra and Trigonometry
SOLVED PROBLEMS
7.19 Simplify the trigonometric function
SOLUTION
1 . cos 2 x sin 2 x
1 TrigReduce Cos[x]2 Sin[x]2 Sec[2 x]
7.20 Factor and simplify: sin2 x cos2 x + cos4 x.
SOLUTION
TrigFactor[Sin[x]2 Cos[x]2 + Cos[x]4] Cos[x]2 7.21 Solve the trigonometric equation 1 2 cos x 2 sin x + 4 sin 2x = 0.
SOLUTION
equation = 1 2 Cos[x] 2 Sin[x] + 4 Sin[2 x] 0 Solve[equation, x]
Solve ifun : Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
1 1 13 1 1 13 1 1 x ArcCos 8 + 8 4 2(9 13) , x ArcCos 8 + 8 + 4 2(9 13) , 1 1 13 13 9 9 x ArcCos 8 1 13 4 8 + 8 , x ArcCos 8 1 13 + 4 8 + 8 A numerical solution would probably be more useful. % //N {{x 1.40492}, {x 0.165873}, {x 2.83487}, {x 1.26407}}
SOLUTION
cos x + tan x . 1 + sin x
Together Cos[x] + Tan[x], Trig True 1 + Sin[x]
1 x Sin x Cos x + Sin x Cos 2 2 2 2
Sometimes you have to apply two or more trig commands to simplify completely.
TrigReduce[%] Sec[x] 7.23 Combine and simplify:
SOLUTION
sinh x cosh x + cosh x sinh x cosh x + sinh x
Sinh[x] Cosh[x] , Trig True Together + o Cosh[x] Sinh[x] Cosh[x] + Sinh[x] Cosh[2 x]
Algebra and Trigonometry
7.24 Construct a table of multiple angle formulas for sin nx and cos nx, n = 2, 3, 4, and 5.
SOLUTION
trigtable = Table[{n, TrigExpand[Sin[n x]], TrigExpand[Cos[n x]]}, {n, 2, 5}]; TableForm[trigtable, TableHeadings {None, {"n", " sin nx", " cos nx"}}]
n 2 3 4 5 sin nx 2 Cos[x]Sin[x] 3 Cos[x]2 Sin[x] Sin[x]3 4 Cos[x] Sin[x] 4 Cos[x]Sin[x]
cos nx Cos[x]2 Sin[x]2 Cos[x]3 3 Cos[x] Sin[x]2 Cos[x]4 6 Cos[x]2 Sin[x]2 + Sin[x]4 Cos[x]5 10 Cos[x]3 Sin[x]2 + 5 Cos[x] Sin[x]4
4 2 3 5 5 Cos[x] Sin[x] 10 Cos[x] Sin[x] + Sin[x]
7.25 Construct a table of linear trig formulas for sinn x and cosn x, n = 2, 3, 4, and 5.
SOLUTION
trigtable = Table[{n, TrigReduce[Sin[x]n], TrigReduce[Cos[x]n]}, {n, 2, 5}]; TableForm[trigtable, TableHeadings {None, {"n", " sinn x", " cosn x"}}]
n 2 3 4 5 sinn x 1(1 Cos[2x]) 2 1 (3 Sin[x] Sin[3x]) 4 1 (3 4 Cos[2x]+ Cos[4x]) 8 1 (10 Sin[x] 5 Sin[3x]+ Sin[5x]) 16 Cosn x 1(1+ Cos[2x]) 2 1 (3 Cos[x]+ Cos[3x]) 4 1 (3 + 4Cos[2x]+ Cos[4x]) 8 1 (10Cos[x]+ 5 Cos[3x]+ Cos[5x]) 16
7.26 Express ex + y in terms of hyperbolic functions and expand.
SOLUTION
ExpToTrig[Ex + y] Cosh[x + y] + Sinh[x + y] TrigExpand[%] Cosh[x]Cosh[y] + Cosh[y]Sinh[x] + Cosh[x]Sinh[y] + Sinh[x]Sinh[y]
7.27 Express sinh 1x and tanh 1x in logarithmic form.
SOLUTION
TrigToExp[ArcSinh[x]]
Log [x + 1 + x2 ]
TrigToExp[ArcTanh[x]]
1 Log [1 x ] + 1 Log [1 + x ] 2 2
7.28 Use Manipulate to control the graph of f(x) = a sin (b x + c), 0 x < 2 , with controls for a, b, and c varying between 1 and 10. Move the sliders and observe the affect upon the graph.
SOLUTION
Manipulate[Plot[a Sin[b x + c], {x, 0, 2 o}, PlotRange { 10, 10}], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}]
Algebra and Trigonometry
a b c
7.4 The Art of Simplification
There are many different ways to write any particular algebraic or trigonometric expression. Obviously one person s interpretation of simple may not agree with another s. For example, in dealing with rational functions, (x + 3)2 may be preferable to x2 + 6 x + 9, but when manipulating polynomials, the latter is clearly more desirable. As you have seen from reading this chapter, Mathematica offers a variety of commands that allow full control of how an expression will appear. With practice, you will learn to use these commands to reshape appearances to suit your needs. As a step in the direction toward simplification, Mathematica offers two commands that can be used to simplify complex structures.
Simplify[expression] performs a sequence of transformations on expression and returns the simplest form it finds. FullSimplify[expression] tries a wider range of transformations on expression including elementary and special functions and returns the simplest form it finds.
Simplify tries expanding, factoring, and other standard mathematical transformations to reduce the complexity of expression. Because of its general nature, Simplify tends to be quite slow in comparison to more direct instructions. FullSimplify always produces an expression at least as simple as Simplify, but may take somewhat longer. You can specify a time limitation (in seconds) with the option TimeConstraint. The default for Simplify is TimeConstraint 300 and for FullSimplify, TimeConstraint Infinity. For both commands, Trig True is the default for trigonometric evaluation.