what is the arc length in the first quadrant between the positive horizonal axis and the point (4,3)

Question

A circle with its center at the origin in a rectangular coordinate system passes through the point (4,3).

Answer 1

A circle with its center at the origin in a rectangular coordinate system passes through the point (4,3).

What is the arc length in the first quadrant between the positive horizontal axis and the point (4,3)

Label the origin as O and the point (4,3) as P.

Drop a perpendicular from P to the x-axis. Label this point, (4,0), as N.

Then NP = 3, and by Pythagoras, OP^2 = ON^2 + NP^2 = 4^2 + 3^2 = 25

i.e. OP = 5

Call the angle PON as θ.

Then tan θ = PN/ON = 3/4 = 0.75

So, θ = 0.6435 rad

Let the circle cut the x-axis at M.

The length of the arc MP is given by s = rθ, where r = OP = 5.

Hence, s = 5*0.6435 = 3.2175

Answer: arc length = 3.2175