I need help finding the angle of a right sided triangle.

Remember:

Some Old Hags,         Sin=Opp/Hyp

Can't Always Hide,     Cos=Adj/Hyp

Their Old Age.           Tan=Opp/Adj

You have adj and hyp so the ratio is Cos.

Now if you want to find an angle you must use inverse trig. That is  $$Cos^{-1}$$  but the Web2 site calc does not have that.  Arc tan is exactly the same thing!  So use acos  

acos(adj/hyp) = acos(13/23)

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{13}}}{{\mathtt{23}}}}\right)} = {\mathtt{55.582\: \!611\: \!289\: \!558^{\circ}}}$$

You would need to use cos theta since cos theta = adjacent/hypotenuse

cos theta = 13/23

cos^-1(13/23) = theta

theta = 55.6^o.

    For me the easest way to find this angle (or either angle of a right triangle) is using square root. In this case 23 squared minus 13 squared = 360. Now find the square root of 360. = 18.97366596 this is the length of side oposite. Now divide 18.97366596 by 23 =.824941998 = the sine of the angle you are looking for. inverse .824941998 = 55.58261129 = the decimal degrees of the angle looking for. Inverse 55.58261129 =55. degree 34 minutes 57.4 seconds. I may not know very much about formules, but I put this to practical use for over 41 years as a Tool & Die Maker (aska machinest) without any problem. If you use a little common sence and not clutter your mind trying to remember to many ways to work a simple right angle (nothing wrong knowing other ways),  Just remember Square and Square root, Tangent and Sine, keep it simple. If you get these down pat you can keep up with the best.   Please dont come down to hard on me for not knowing all the fancy ways, just keeping it simple.  This is Rustynut "just getting by"