simplify trigonometric expressions
literal name; trig
The simplify(expr,trig) calling sequence simplifies trigonometric expressions by applying the trigonometric identities sin⁡x2+cos⁡x2=1 and cosh⁡x2−sinh⁡x2=1.
If the input is a polynomial in sin⁡x and cos⁡x then simplify/trig factors out powers of sin⁡x and cos⁡x and apply the identity sin⁡x2+cos⁡x2=1 to what is left so that the degree of what is left in sin⁡x is at most 1. Thus the result is of the form:
where A and B are polynomials in cos⁡x. If the input is a polynomial in sinh⁡x and cosh⁡x then simplify/trig yields a similar result using the identity cosh⁡x2−sinh⁡x2=1.
To apply the identity to reduce the polynomial so that the degree in sin⁡x is at most 1, use the command
If the input involves multiple angles that are integer multiples of each other, for example, sin⁡x, sin2⁢x, and cos⁡x2 then the trigonometric functions are expressed in terms of a common angle, in this case x2.
If the input is a rational expression in sin⁡x and cos⁡x 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 sin⁡x and cos⁡x) 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 AB⁢sin⁡x+CD for polynomials A, B, C, and D in cos⁡x because this form usually leads to a result that is larger in total degree.
f ≔ sin⁡x⁢cos⁡x2−cos⁡x3+cos⁡x
r ≔ 1−cos⁡x2+sin⁡x⁢cos⁡xsin⁡x⁢cos⁡x+cos⁡x2
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