This is the AMBIGUOUS case SSA, (side-side-angle) which may have 0 solutions,
1 solution, or 2 solutions. If it has 2 solutions we call them triangle ABC
with sides a,b,c, and triangle A'B'C' with sides a',b',c'.

I'll tell you in advance that there are two solutions to this problem, 
The first two figures below show the two solutions, and the third
figure below shows them put together with triangle A'B'C' inside of
triangle ABC, the blue arc shows that a and a' both equal 9 in length:




First solution               Second solution (maybe)

A = 60°                      A' = 60°
B =                          B' = 
C =                          C' = 
a =  9                       a' =  9  
b =                          b' =  
c = 10                       c' = 10


We start with what you had:

 = 

 = 

9·sin(C) = 10·sin(60°)

sin(C) = 

sin(C) = .9622504486  <--- If this had been greater than 1, there
                           whould have been no solution.  If this had 
                           been exactly 1, there would have been 1 right
                           triangle angle solution.  But since it is less \
                           than 1, we can tell that there is either 1 or 2
                           solutions.  
                
If you use your calculator with the inverse sine,
you get 

C = 74.20683095°.

That is a correct value for angle C.  However there is another 
possible angle with that same sine, which is a second quadrant 
angle, and it is found by subtracting 74.20683095° from 180°.
We'll call it C':

C' = 180° - 74.20683095° = 105.793169°

So we put those two values in:

First solution               Second solution (maybe)

A = 60°                      A' = 60°
B =                          B' = 
C = 74.20683095°.            C' = 105.793169°
a =  9                       a' =  9  
b =                          b' =  
c = 10                       c' = 10  

To find out whether there are 2 solutions or only 1,
we calculate the angles B and B', and see if both
are possibilities:

We calculate B by using 180°-A-C = 180°-60°-74.20683095° = 45.79316905°

We calculate B' by using 180°-A'-C' = 180°-60°-105.793169° = 14.20683095°

Since B' came out positive, we know that there are two solutions,
[If B' had come out negative, there would have been only 1 solution].

First solution               Second solution

A = 60°                      A' = 60°
B = 45.79316905°             B' = 14.20683095°
C = 74.20683095°.            C' = 105.793169°
a =  9                       a' =  9  
b =                          b' =  
c = 10                       c' = 10 

Now we just have to calculate sides b and b'

To calculate side b:

 = 

 =  

b·sin(60°) = 9·sin(45.79316905°)  

b = 

b = 7.449489743

---

To calculate side b':

 = 

 =  

b'·sin(60°) = 9·sin(45.79316905°)  

b' = 

b' = 2.550510257

So we end up with:

First solution               Second solution

A = 60°                      A' = 60°
B = 45.79316905°             B' = 14.20683095°
C = 74.20683095°             C' = 105.793169°
a =  9                       a' =  9  
b = 7.449489743              b' = 2.550510257  
c = 10                       c' = 10

Edwin