Find to at least three decimal places the coordinates of the

Find (to at least three decimal places) the coordinates of the point p in the figure where the tangent line to y = 2cos(5x) passes through the origin. The coordinates of P are (, )

Solution

dy/dx=-10 sin(5x)

Let, P=(x0,y0)

So,

(y-y0)=-10 sin(5x0)(x-x0)

y=-10 sin(5x0)x+10 x0 sin(5x0)+y0

Since line passes through origin

10 x0 sin(5x0)+y0=0

10 x0 sin(5x0)+2 cos(5x0)=0

5x0=-cot(5x0)

Solving this using Wolfram Alpha and using the graph which shows: 0<x0<2pi/5

2pi/5~1.25

x0~0.560

y0~1.998