[Ezra Halleck]

3.6: Find the coordinates of all points on the curve , , where the tangent line is parallel to the line . Draw a quick sketch of curve, original line and the tangent lines.

[Ezra Halleck]

1. slope of line is the coefficient of x which is (after distributing the 2) -2/sqrt(3).
2. Derivative of cscx is -cscx cotx= -cosx/sin^2x
3. Set derivative to our value of m: -cosx/sin^2x=-2/sqrt(3)
4. clear denominators and multiply by -1 to get sqrt(3)cosx=2sin^2x.
5. replace sin^2x w/ 1 – cos^2x, distribute 2 and bring RHS to left to get 2cos^2x+sqrt(3)cosx-2=0, quadratic in cos(x).
6. Using the quadratic formula, we get cosx=(-sqrt(3)+ or – sqrt(3-4(2)(-2)))/4=(-sqrt(3)+ or – sqrt(19))/4.
7. Since |cosx|<=1, the minus gives no solution so cosx=(-sqrt(3) + sqrt(19))/4
8. Use a calculator and cos^-1 to get x in QII: 0.8543. Realizing that there is also a solution in QIII that is symmetric with respect to the x-axis, we subtract QII solution from 2pi: 2pi- 0.8543=5.4288.
In the attached file, graph of original line (blue), the function (blue) and the tangent lines (green and red, respectively).