trigonometric - Maple Help

simplify/trig

simplify trigonometric expressions

 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  ${\mathrm{sin}\left(x\right)}^{2}+{\mathrm{cos}\left(x\right)}^{2}=1$ and ${\mathrm{cosh}\left(x\right)}^{2}-{\mathrm{sinh}\left(x\right)}^{2}=1$.
 • If the input is a polynomial in $\mathrm{sin}\left(x\right)$ and $\mathrm{cos}\left(x\right)$ then simplify/trig factors out powers of $\mathrm{sin}\left(x\right)$ and $\mathrm{cos}\left(x\right)$ and apply the identity ${\mathrm{sin}\left(x\right)}^{2}+{\mathrm{cos}\left(x\right)}^{2}=1$ to what is left so that the degree of what is left in $\mathrm{sin}\left(x\right)$ is at most 1. Thus the result is of the form:

${\mathrm{sin}\left(x\right)}^{i}{\mathrm{cos}\left(x\right)}^{j}\left(A\mathrm{sin}\left(x\right)+B\right)$

 where $A$ and $B$ are polynomials in $\mathrm{cos}\left(x\right)$. If the input is a polynomial in $\mathrm{sinh}\left(x\right)$ and $\mathrm{cosh}\left(x\right)$ then simplify/trig yields a similar result using the identity ${\mathrm{cosh}\left(x\right)}^{2}-{\mathrm{sinh}\left(x\right)}^{2}=1$.
 • To apply the identity to reduce the polynomial so that the degree in $\mathrm{sin}\left(x\right)$ is at most 1, use the command

$\mathrm{simplify}\left(\mathrm{expr},\left\{{\mathrm{sin}\left(x\right)}^{2}+{\mathrm{cos}\left(x\right)}^{2}-1\right\},\left[\mathrm{sin}\left(x\right)\right]\right)$

 • If the input involves multiple angles that are integer multiples of each other, for example, $\mathrm{sin}\left(x\right)$, $\mathrm{sin}\left(2x\right)$, and $\mathrm{cos}\left(\frac{x}{2}\right)$ then the trigonometric functions are expressed in terms of a common angle, in this case $\frac{x}{2}$.
 • If the input is a rational expression in $\mathrm{sin}\left(x\right)$ and $\mathrm{cos}\left(x\right)$ then an algorithm is used to put it in the form $\frac{N}{\mathrm{D}}$ and reduce $N$ and $\mathrm{D}$ to lowest terms such that the total degree of the numerator $N$ (in $\mathrm{sin}\left(x\right)$ and $\mathrm{cos}\left(x\right)$) plus the total degree of the denominator is minimized.  In particular, any common factor between $N$ and $\mathrm{D}$ has been cancelled out.
 Note: Maple does not rationalize the denominator, that is, write the expression in the form $\frac{A}{B}\mathrm{sin}\left(x\right)+\frac{C}{\mathrm{D}}$ for polynomials $A$, $B$, $C$, and $\mathrm{D}$ in $\mathrm{cos}\left(x\right)$ because this form usually leads to a result that is larger in total degree.

Examples

 > $\mathrm{simplify}\left({\mathrm{sin}\left(x\right)}^{2}+{\mathrm{cos}\left(x\right)}^{2},\mathrm{trig}\right)$
 ${1}$ (1)
 > $\mathrm{simplify}\left({\mathrm{cosh}\left(x\right)}^{2}-{\mathrm{sinh}\left(x\right)}^{2},\mathrm{trig}\right)$
 ${1}$ (2)
 > $\mathrm{simplify}\left(\mathrm{cos}\left(2x\right)+{\mathrm{sin}\left(x\right)}^{2},\mathrm{trig}\right)$
 ${{\mathrm{cos}}{}\left({x}\right)}^{{2}}$ (3)
 > $f≔\mathrm{sin}\left(x\right){\mathrm{cos}\left(x\right)}^{2}-{\mathrm{cos}\left(x\right)}^{3}+\mathrm{cos}\left(x\right)$
 ${f}{≔}{\mathrm{sin}}{}\left({x}\right){}{{\mathrm{cos}}{}\left({x}\right)}^{{2}}{-}{{\mathrm{cos}}{}\left({x}\right)}^{{3}}{+}{\mathrm{cos}}{}\left({x}\right)$ (4)
 > $\mathrm{simplify}\left(f,\mathrm{trig}\right)$
 ${\mathrm{sin}}{}\left({x}\right){}{\mathrm{cos}}{}\left({x}\right){}\left({\mathrm{cos}}{}\left({x}\right){+}{\mathrm{sin}}{}\left({x}\right)\right)$ (5)
 > $r≔\frac{1-{\mathrm{cos}\left(x\right)}^{2}+\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)}{\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)+{\mathrm{cos}\left(x\right)}^{2}}$
 ${r}{≔}\frac{{\mathrm{sin}}{}\left({x}\right){}{\mathrm{cos}}{}\left({x}\right){-}{{\mathrm{cos}}{}\left({x}\right)}^{{2}}{+}{1}}{{\mathrm{sin}}{}\left({x}\right){}{\mathrm{cos}}{}\left({x}\right){+}{{\mathrm{cos}}{}\left({x}\right)}^{{2}}}$ (6)
 > $\mathrm{simplify}\left(r,\mathrm{trig}\right)$
 $\frac{{\mathrm{sin}}{}\left({x}\right)}{{\mathrm{cos}}{}\left({x}\right)}$ (7)
 > $\mathrm{simplify}\left(\left[{\mathrm{sin}\left(x\right)}^{3},{\mathrm{sin}\left(x\right)}^{4}\right],\mathrm{trig}\right)$
 $\left[{{\mathrm{sin}}{}\left({x}\right)}^{{3}}{,}{{\mathrm{sin}}{}\left({x}\right)}^{{4}}\right]$ (8)
 > $\mathrm{simplify}\left(\left[{\mathrm{sin}\left(x\right)}^{3},{\mathrm{sin}\left(x\right)}^{4}\right],\left\{{\mathrm{sin}\left(x\right)}^{2}+{\mathrm{cos}\left(x\right)}^{2}-1\right\},\left[\mathrm{sin}\left(x\right)\right]\right)$
 $\left[{-}{\mathrm{sin}}{}\left({x}\right){}{{\mathrm{cos}}{}\left({x}\right)}^{{2}}{+}{\mathrm{sin}}{}\left({x}\right){,}{{\mathrm{cos}}{}\left({x}\right)}^{{4}}{-}{2}{}{{\mathrm{cos}}{}\left({x}\right)}^{{2}}{+}{1}\right]$ (9)
 > $\mathrm{simplify}\left(\mathrm{sin}\left(4x\right)\mathrm{cos}\left(2x\right),\mathrm{trig}\right)$
 ${2}{}{\mathrm{sin}}{}\left({2}{}{x}\right){}{{\mathrm{cos}}{}\left({2}{}{x}\right)}^{{2}}$ (10)
 > $\mathrm{expand}\left(\mathrm{sin}\left(4x\right)\right)$
 ${8}{}{\mathrm{sin}}{}\left({x}\right){}{{\mathrm{cos}}{}\left({x}\right)}^{{3}}{-}{4}{}{\mathrm{sin}}{}\left({x}\right){}{\mathrm{cos}}{}\left({x}\right)$ (11)
 > $\mathrm{combine}\left({\mathrm{sin}\left(x\right)}^{4},\mathrm{trig}\right)$
 $\frac{{3}}{{8}}{+}\frac{{\mathrm{cos}}{}\left({4}{}{x}\right)}{{8}}{-}\frac{{\mathrm{cos}}{}\left({2}{}{x}\right)}{{2}}$ (12)