Help with trigonometry needed

Question

I am trying to learn mechanics, but my math skills are quite poor. My text book tells me that we have two equations: $$-N*\sin(\alpha)+S*\sin(\beta)=0$$ $$-G+N*\cos(\alpha)+S*\cos(\beta)=0$$ And after we have done the math we should find that:
$$N=G*\frac{\sin(\beta)}{\sin(\alpha +\beta)}$$ and $$S=G*\frac{\sin(\alpha)}{\sin(\alpha +\beta)}$$ Can someone please tell me how did we get from one to the other?

Answer 1

$N$ and $S$ are the two unknowns, and you have two equations. The simplest method (at least in this case) is to use one equation to solve for one of the unknowns in terms of the other, and then substitute the result in the other equation. You will then have one equation in one unknown which you can presumably solve. Then you substitute that solution back.

Here, we'll use the first equation to express $N$ in terms of $S$:

$$\begin{align} -N \sin \alpha + S \sin \beta &= 0 \implies\\ N &= S \frac{\sin \beta}{\sin \alpha} \tag{1} \end{align}$$

We then substitute for $N$ in the second equation:

$$ \begin{align} -G + N\cos \alpha + S \cos \beta &= 0 \implies\\ -G + S\frac{\cos \alpha \sin \beta}{\sin \alpha} + S \cos \beta &= 0 \implies\\ S\left(\frac{\cos \alpha \sin \beta}{\sin \alpha} + \cos \beta \right) &= G \implies\\ S\left(\frac{\cos \alpha \sin \beta + \sin \alpha\ cos \beta}{\sin \alpha}\right) &= G \implies \\ S &= G \frac{\sin \alpha}{\cos \alpha \sin \beta + \sin \alpha \cos \beta} \implies\\ S &= G \frac{\sin \alpha}{\sin(\alpha + \beta)} \tag{2} \end{align}$$

The last equation follows from the trigonometric identity for $\sin(\alpha + \beta)$:

$$ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$

We now substitute equation (2) into equation (1):

$$\require{cancel}\begin{align} N &= S \frac{\sin \beta}{\sin \alpha} \implies\\ N &= G \frac{\cancel{\sin \alpha}}{\sin(\alpha + \beta)} \frac{\sin \beta}{\cancel{\sin \alpha}} \implies\\ N &= G \frac{\sin \beta}{\sin(\alpha + \beta)} \tag{3} \end{align} $$

and we are done: equations (2) and (3) express the unknowns $N$ and $S$ in terms of the (presumably known) $G$ and the angles $\alpha$ and $\beta$.