Let's draw a diagram of the problem:
The Law of Sines tells us that
[tex]\frac{x}{\sin X}=\frac{v}{\sin V}=\frac{w}{\sin W}\text{.}[/tex]
Since we don't have information about either v or ∠V, we can reduce the equation above to
[tex]\frac{x}{\sin X}=\frac{w}{\sin W}\text{.}[/tex]
Using the values we were given, we get:
[tex]\frac{5.3}{\sin X}=\frac{7.3}{\sin (37)}\text{.}[/tex]
We are interested in finding the possible values of ∠X, so let's rewrite this equation. First, let's multiply both sides by sinX:
[tex]5.3=\frac{7.3\cdot\sin X}{\sin (37)}\text{.}[/tex]
Next, let's multiply both sides by sin(37):
[tex]5.3\cdot\sin (37)=7.3\cdot\sin X\text{.}[/tex]
Now let's divide both sides by 7.3:
[tex]\frac{5.3\cdot\sin(37)}{7.3}=\sin X,[/tex]
Using a calculator or online resource, we can calculate the left side of the equation:
[tex]\frac{5.3\cdot\sin(37)}{7.3}\approx\frac{5.3\cdot0.6018}{7.3}\approx0.4369.[/tex]
So our equation becomes
[tex]\sin X=0.4369.[/tex]
Using a calculator or online resource, we can obtain the value of the inverse function of sine:
[tex]X=\arcsin (0.4369),[/tex]
The solutions to this equation are given by
[tex]X=\arcsin (0.4369)+360h[/tex]
and
[tex]X=-\arcsin (0.4369)+360h+180,[/tex]
where h can be any integer number.
The only solutions we'll find that make sense are when X<180°, since X is an angle in a triangle, so when h=0, we get, from the first equation:
[tex]X=\arcsin (0.4369)=25.91,[/tex]
and from the second:
[tex]X=-25.91+180=154.09.[/tex]
Any other value of h will make X greater than 180° or lower than 0°, which doesn't make sense since again, X is an angle in a triangle.
Thus, the only possible solution, round to a tenth of a degree is ∠X=25.9°.
