Question 1087083
A*sin(X)+B*sin(X+U)=C*sin(X+W), with C=sqrt(A^2+B^2+2*A*B*cos(U)) 
and tan(W)=(B*sin(U))/(A+B*cos(U)).
<pre>
We are given that  tan(W)=(B*sin(U))/(A+B*cos(U)).

Since {{{TANGENT=OPPOSITE/ADJACENT}}} and

{{{tan^""(W)=(B*sin^""(U))/(A+B*cos^""(U))}}},

we can draw a right triangle with B*sin(U) as the
opposite side of angle W and A+B*cos(U) as the
adjacent side of W:

{{{drawing(400,2000/9,-1,8,-1,4,triangle(0,0,5,0,5,3),
arc(0,0,3,-3,0,31), locate(2.5,2,"??"),
locate(1,.5,W), locate(1.5,0,A+B*cos^""(U)), locate(5.1,1.6,B*sin^""(U)) )}}} 

We calculate the hypotenuse:

{{{HYPOTENUSE^2}}}{{{""=""}}}{{{ADJACENT^2+OPPOSITE^2}}}

{{{HYPOTENUSE^2}}}{{{""=""}}}{{{(A+B*cos^""(U)^"")^2+(B*sin^""(U)^"")^2}}}

{{{HYPOTENUSE^2}}}{{{""=""}}}{{{A^2+2AB*cos^""(U)+B^2*cos^2(U)+B^2*sin^2(U)}}}

{{{HYPOTENUSE^2}}}{{{""=""}}}{{{A^2+2AB*cos^""(U)+B^2(cos^2(U)^""+sin^2(U))}}}

{{{HYPOTENUSE^2}}}{{{""=""}}}{{{A^2+2AB*cos^""(U)+B^2(1^""^"")}}}

{{{HYPOTENUSE^2}}}{{{""=""}}}{{{A^2+2AB*cos^""(U)+B^2}}}

{{{HYPOTENUSE}}}{{{""=""}}}{{{sqrt(A^2+2AB*cos^""(U)+B^2)}}}

Notice that this result is the exact value that was given for C. So

{{{HYPOTENUSE}}}{{{""=""}}}{{{C}}}

{{{drawing(400,2000/9,-1,8,-1,4,triangle(0,0,5,0,5,3),
arc(0,0,3,-3,0,31), locate(2.5,2,C),
locate(1,.5,W), locate(1.5,0,A+B*cos^""(U)), locate(5.1,1.6,B*sin^""(U)) )}}}

Let's start with right side of what we have to prove:

{{{C*sin^""(X+W)}}}{{{""=""}}}{{{C(sin^""(X)cos^""(W)+cos^""(X)sin^""(W)^"")}}}
{{{C*sin^""(X+W)}}}{{{""=""}}}{{{C*sin^""(X)cos^""(W)+C*cos^""(X)sin^""(W)^""}}}

Since {{{COSINE=ADJACENT/HYPOTENUSE}}} and {{{SINE=OPPOSITE/HYPOTENUSE}}},
we use the right triangle above to substitute for cos(W) and sin(W)

{{{C*sin^""(X+W)}}}{{{""=""}}}{{{C*sin^""(X)((A+B*cos^""(U))/C)+C*cos^""(X)((B*sin^""(U)^"")/C)}}}

{{{C*sin^""(X+W)}}}{{{""=""}}}{{{cross(C)*sin^""(X)((A+B*cos^""(U))/cross(C))+cross(C)*cos^""(X)((B*sin^""(U)^"")/cross(C))}}}

{{{C*sin^""(X+W)}}}{{{""=""}}}{{{sin^""(X)(A+B*cos^""(U)^"")+cos^""(X)(B*sin^""(U)^"")}}}

{{{C*sin^""(X+W)}}}{{{""=""}}}{{{A*sin^""(X)+B*sin^""(X)cos^""(U)^""+B*cos^""(X)*sin^""(U)^""}}}

{{{C*sin^""(X+W)}}}{{{""=""}}}{{{A*sin^""(X)+B(sin^""(X)^""cos^""(U)^""+cos^""(X)*sin^""(U)^"")}}}

{{{C*sin^""(X+W)}}}{{{""=""}}}{{{A*sin^""(X)+B*cos^""(X+U)^""}}}

Edwin</pre>