Hello, blame_canada100!
I found a solution.
But there may be (there should be) a more elegant one.
The measures of the angles between the resultant
. . and two applied forces are 65° and 42°.
If the magnitude of the resultant is 24 pounds,
. . find to the nearest pound the magnitude of the smaller force. Code:
R S
* *
|* * |*
| * * | *
| * b 24 * | *
| * * | *
| * * | *
| * 65° * | *
| * * 42° | *
* - - - * - - - - - - - * - - - *
P a Q
The two vectors are: .$\displaystyle \begin{array}{ccccc}\vec u &=& \overrightarrow{PQ} &=& \langle a,\:0\rangle\qquad\qquad \\ \vec v &=& \overrightarrow{PR} &=& \langle b\cos107^o,b\sin107^o\rangle \end{array}$
. . .The resultant is: .$\displaystyle \vec w \;=\;\overrightarrow{PS} \;=\;\langle24\cos42^o, 24\sin42^o\rangle$
We have: .$\displaystyle \vec u + \vec v \:=\:\vec w \quad\Rightarrow\quad \langle a,0\rangle + \langle b\cos107^o,b\sin107^o\rangle \:=\:\langle24\cos42^o, 24\sin42^o\rangle $
Hence: .$\displaystyle \begin{array}{ccc}a + b\cos107^o &=& 24\cos42^o \\ b\sin107^o &=& 24\sin42^o \end{array}$
Solve the system for $\displaystyle a$ and $\displaystyle b$ and see which is smaller.