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simplify/trig

simplify trigonometric expressions

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

simplify(expr, trig)

Parameters

expr

-

any expression

trig

-

literal name; trig

Description

• 

The simplify(expr,trig) calling sequence simplifies trigonometric expressions by applying the trigonometric identities  sinx2+cosx2=1 and coshx2sinhx2=1.

• 

If the input is a polynomial in sinx and cosx then simplify/trig factors out powers of sinx and cosx and apply the identity sinx2+cosx2=1 to what is left so that the degree of what is left in sinx is at most 1. Thus the result is of the form:

sinxicosxjAsinx+B

  

where A and B are polynomials in cosx. If the input is a polynomial in sinhx and coshx then simplify/trig yields a similar result using the identity coshx2sinhx2=1.

• 

To apply the identity to reduce the polynomial so that the degree in sinx is at most 1, use the command

simplifyexpr,sinx2+cosx21,sinx

• 

If the input involves multiple angles that are integer multiples of each other, for example, sinx, sin2x, and cos12x then the trigonometric functions are expressed in terms of a common angle, in this case 12x.

• 

If the input is a rational expression in sinx and cosx then an algorithm is used to put it in the form ND and reduce N and D to lowest terms such that the total degree of the numerator N (in sinx and cosx) plus the total degree of the denominator is minimized.  In particular, any common factor between N and D has been cancelled out.

  

Note: Maple does not rationalize the denominator, that is, write the expression in the form ABsinx+CD for polynomials A, B, C, and D in cosx because this form usually leads to a result that is larger in total degree.

Examples

simplifysinx2+cosx2,trig

1

(1)

simplifycoshx2sinhx2,trig

1

(2)

simplifycos2x+sinx2,trig

cosx2

(3)

fsinxcosx2cosx3+cosx

f:=sinxcosx2cosx3+cosx

(4)

simplifyf,trig

sinxcosxcosx+sinx

(5)

r1cosx2+sinxcosxsinxcosx+cosx2

r:=sinxcosxcosx2+1sinxcosx+cosx2

(6)

simplifyr,trig

sinxcosx

(7)

simplifysinx3,sinx4,trig

sinx3,sinx4

(8)

simplifysinx3,sinx4,sinx2+cosx21,sinx

cosx2+1sinx,cosx42cosx2+1

(9)

simplifysin4xcos2x,trig

2sin2xcos2x2

(10)

expandsin4x

8sinxcosx34sinxcosx

(11)

combinesinx4,trig

38+18cos4x12cos2x

(12)

See Also

combine

expand

simplify

 


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