first off, it really helps to make a sketch when solving problems using vectors.12. A force of 176 lbs. Makes an angle of 78°50 with a second force. The resultant of the two forces makes an angle of 41°10 with the first force. Find the magnitude of the second force and the resultant.

use a coordinate system to your advantage ... let the first force, , point along the (+) x-axis.

let the second force have magnitude ... its direction will be 78°50 relative to the (+) x-axis

let the resultant have magnitude ... its direction will be 41°10 relative to the (+) x-axis

in the x direction ...

in the y-direction ...

using the second equation, solve for ...

substitute this expression for in the first equation ...

I get and

I expect round-off error on the part of whoever came up with the provided solutions ... I did not round off in any step of my calculations.