Find common hypotenuse for 2 triangles: boiled down to Y/sin(x) = Z/sin(A-x)

[Paperfrog]

I try to find a common hypotenuse of two triangles and have (hopefully right) boiled it down to this formula where I need to isolate the angle x.

Y/sin(x) = Z/sin(A-x)


Any help would be appreciated!

 

[lev888]

sin(A+B)=sin A cos B + cos A sin B
sin(A-B)=sin A cos B - cos A sin B
cos(A+B)=cos A cos B - sin A sin B
cos(A-B)=cos A cos B + sin A sin B

 

[Steven G]

Please post the original problem IN FULL...
What you posted is useless.

The student just wants to solve for x, which is possible. They have confidence in their prior work:|

 

[HallsofIvy]

I try to find a common hypotenuse of two triangles and have (hopefully right) boiled it down to this formula where I need to isolate the angle x.

Y/sin(x) = Z/sin(A-x)

So Y sin(A- x)= Z sin(x).

Using sin(A-B)=sin A cos B - cos A sin B, as lev888 posted, sin(A- x)= sin(A)cos(x)- cos(A)sin(x) so that equation becomes Ysin(A)cos(x)- Ycos(A)sin(x)= Z sin(x).

Then Ysin(A)cos(x)= (Ycos(A)+ Z) sin(x)

tan(x)= sin(x)/cos(x)= (Ycos(A)+ Z)/Ysin(A).

Any help would be appreciated!