Given the right triangle mlk and pln below ; pl=15, kn=4 and cos(l)= (4/5). find the length of km

For the angle PLN, the hypotenuse PL = 15, the cosine of it is 4/5,
what about the length of NL? well, NL would be the "adjacent" side
of the angle PLN, so, recall your SOH CAH TOA



which of those fellows, give us only
the angle
adjacent side
and
hypotenuse?

well, is Ms Cosine, so let's bother her


now, that we know what NL is,  what is ML?

notice, the angle MLK is really the same shared angle PLN
so the cosine of that is the same 4/5

so let us use cosine again, now for the bigger triangle though


so.. now we know what the  sides of KL and ML are,
but what about MK?   well
notice, is a right-triangle,
you have the hypotenuse ML,
you also have the adjacent side, KL,

to find the opposite side of MK, simply use the pythagorean theorem