Let O be the origin and P be a point in the fourth quadrant on the x-y plane. Let 270<z<360 be the angle formed by OP with the positive x-axis. Similarly Q is a point in the fourth quadrant on the x-y plane where |OQ|=|OP| and the angle formed by OQ and the x-axis is 7z . For what value of z do the segments OP and OQ coincide?

Question

Let O be the origin and P be a point in the fourth quadrant on the x-y plane. Let 270<z<360 be the angle formed by OP with the positive x-axis. Similarly Q is a point in the fourth quadrant on the x-y plane where |OQ|=|OP| and the angle formed by OQ and the x-axis is 7z . For what value of z do the segments OP and OQ coincide?

Answer 1

we need 7z = z+n*2pi

6z = n*2pi
3z = n*pi
z = n * pi/3

Now, we know that z in is QIV, so it must be 5pi/3.

check:
7*5pi/3 = 35pi/3 = 10pi + 5pi/3

so P and Q coincide if z = 5pi/3