Question

A traffic sign is hanging from two wires. If the weight of the sign is 35 N, what are the tensions in the two wires as shown below? (# in degrees)
---------------------
\ 56 45/
\ /
\ /
\ /

Answer 1

As you can see, it is virtually impossible to draw a figure in this format.

I assume you can sketch a position diagram and a vector diagram of the problem.
My vector diagram has a vertical line of 35 N
the top angle is 34° (90-56)°
the bottom angle is 45°
and the angle opposite the 35 side is therefore 101°

using the sine law twice
string1/sin45 = 35/sin101
string1 = 35sin45/sin101 = 65.88

string2/sin34 = 35/sin101
string2 = 40.97

check my arithmetic

Answer 2

Reiny, I do not see how your calculation = 65.88, I get 25.2.

Also, I do not understand your diagram.
It looks like this

-------------------------------------
\ 56 degree 45 degree /
\ /
\ /
\ /
\ /
\ /
\______________________/
| |
| 35 newtons |
| |
|_____________________|

Answer 3

-------------------------------------
\.56 degree ...........45 degree.../
\................................/
\............................../
\............................/
\........................../
\......................../
\______________________/
|.....................|
|........35 newtons...|
|.....................|
|_____________________|