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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.