Precalculus, Chapter 6, 6.4, Section 6.4, Problem 40

You need to use the formula of dot product to find the angle between two vectors, u = u_x*i + u_y*j, v = v_x*i + v_y*j , such that:
u*v = |u|*|v|*cos(theta)
The angle between the vectors u and v is theta.
cos theta = (u*v)/(|u|*|v|)
First, you need to evaluate the product of the vectors u and v, such that:
u*v = u_x*v_x + u_y*v_y
u*v = cos(pi/4)*cos(pi/2) + sin(pi/4)*sin(pi/2)
u*v = cos(pi/2-pi/4) = sin pi/4 = sqrt2/2
You need to evaluate the magnitudes |u| and |v|, such that:
|u|= sqrt(cos^2(pi/4) + sin^2(pi/4)) => |u|= sqrt(1) =>|u|= 1
|v|= sqrt(cos^2(pi/2) + sin^2(pi/2)) => |v|= sqrt(1) =>|v|= 1
cos theta = (sqrt2/2)/(1*1) => cos theta = sqrt2/2 => theta = pi/4
Hence, the cosine of the angle between the vectors u and v is cos theta = sqrt2/2 , so, theta = pi/4.

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