96.2.1 Convert to sin and cos

Function: c2sin (x) ¶

Function: c2cos (x) ¶

The function c2sin converts the expression \(a\cos x + b\sin x\) to \(r\sin(x+\phi).\)

The function c2cos converts the expression \(a\cos x + b\sin x\) to \(r\cos(x-\phi).\)

Examples:

(%i1) load("trigtools")$
(%i2) c2sin(3*sin(x)+4*cos(x));
                                            4
(%o2)                        5 sin(x + atan(-))
                                            3
(%i3) trigexpand(%),expand;
(%o3)                        3 sin(x) + 4 cos(x)

(%i4) c2cos(3*sin(x)-4*cos(x));
                                             3
(%o4)                       - 5 cos(x + atan(-))
                                             4

(%i5) trigexpand(%),expand;
(%o5)                        3 sin(x) - 4 cos(x)

(%i6) c2sin(sin(x)+cos(x));
                                            %pi
(%o6)                       sqrt(2) sin(x + ---)
                                             4

(%i7) trigexpand(%),expand;
(%o7)                          sin(x) + cos(x)
(%i8) c2cos(sin(x)+cos(x));
                                            %pi
(%o8)                       sqrt(2) cos(x - ---)
                                             4

(%i9) trigexpand(%),expand;
(%o9)                          sin(x) + cos(x)

Example. Solve trigonometric equation

(%i10) eq:3*sin(x)+4*cos(x)=2;
(%o10)                      3 sin(x) + 4 cos(x) = 2

(%i11) plot2d([3*sin(x)+4*cos(x),2],[x,-%pi,%pi]);

(%i12) eq1:c2sin(lhs(eq))=2;
                                           4
(%o35)                      5 sin(x + atan(-)) = 2
                                           3
(%i13) solvetrigwarn:false$
(%i14) solve(eq1)[1]$ x1:rhs(%);
                                    2         4
(%o15)                         asin(-) - atan(-)
                                    5         3
(%i16) float(%), numer;
(%o39)                       - 0.5157783719341241
(%i17) eq2:c2cos(lhs(eq))=2;
                                           3
(%o17)                      5 cos(x - atan(-)) = 2
(%i18) solve(eq2,x)[1]$ x2:rhs(%);
                                    3         2
(%o19)                         atan(-) + acos(-)
                                    4         5
(%i20) float(%), numer;
(%o20)                         1.802780589520693

(%i21) sol:[x1,x2];
                          2         4        3         2
(%o44)              [asin(-) - atan(-), atan(-) + acos(-)]
                          5         3        4         5

Answ.: \(x = x_1 + 2\pi k,\) \(x_1 = \sin^{-1}{2\over 5} - \tan^{-1}{4\over 3},\) or \(x_1 = \tan^{-1}{3\over 4} + \cos^{-1}{2\over 5},\) for \(k\) any integer.