Don't understand part of the answer (Pythagoras, Trigonometry, Triangles)

Then it uses the sum of the angles, multiplied by sin and also SQRT(5^2 +2^2)

Notice that triangle DAP is a right angled triangle. So one can use SOHCAHTOA to say that:

$\sin (\angle DAP) = \dfrac{\text{opp}}{\text{hyp}} = \dfrac{DP}{AD}$

And after a bit of basic rearranging, we can see that $DP = AD \times \sin (\angle DAP)$. So, it's clear that if we can figure out the length AD and the angle DAP, then we can plug those in to the right hand side up there and find DP (which is what you've been asked to find in the first place).

So now the question breaks down into two chunks:

(1) Find Angle DAP
(2) Find AD

Notice that one way to find angle DAP is to find angles BAC and DAC and add them together - so, to answer your first question, this means that angle BAC and angle DAC are found in order to find angle DAP.


Notice the formula we found up there for the length DP. See the similarity?

Careful, by the way, that quantity is NOT "the sum of the angles multiplied by sin" - that is 'the sine of the sum of the angles'. Multiplying something by "sin" is meaningless unless it's sin(of something), as sin in it's own right is not a quantity.