How do you find resultant vector through Pythagorean theorem?

Consider vectors A and B.

A has magnitude s and makes an angle a with the positive direction of the x-axis.

B has magnitude t and makes an angle b with the positive direction of the x-axis.

Then the components of A and B along the x-axis are, respectively, s*cos(a) and t*cos(b).

Thus the total horizontal component is u = s*cos(a) and t*cos(b).

The components of A and B along the y-axis are, respectively, s*sin(a) and t*sin(b).

Thus the total vertical component is v = s*sin(a) and t*sin(b).

The magnitude of the resultant, by Pythagoras, is then sqrt(u2 + v2)