For this problem we have a point given (5,y) and we know that this point is on a terminal side of an angle, we also know that:
[tex]\cos \theta=\frac{5}{13}[/tex]If we know the cos then we can find the sin on this way:
[tex]\sin \theta=\frac{y}{13}[/tex]Then we can apply the following identity from trigonometry:
[tex]\sin ^2\theta+\cos ^2\theta=1[/tex]Using this formula we got:
[tex](\frac{5}{13})^2+(\frac{y}{13})^2=1[/tex]And we can solve for y:
[tex]\frac{y^2}{169}=1-\frac{25}{169}=\frac{144}{169}[/tex]And solving for y we got:
[tex]y=\sqrt{169\cdot\frac{144}{169}}=\sqrt{144}=\pm12[/tex]And the two possible solutions for this case are y=12 and y=-12
