Question

How do you use a graphing utility to approximate the solutions of `csc^2x+0.5cotx-5=0` in the interval `[0,2pi)`?

Answer 1

0.56, 2.56, 3.15, 5.70 and 6.24.

Explanation:

The period for the function is `2pi`.

The general solutions are

`x = 2kpi` + a soluion in the list, `k = +-1, +-2, +-3, ...`

The first graph gives locations. The other graphs narrow down to

near-linearity, near the solutions.

These approximations can be used as starters, for numerical

iterative methods that would generate quite many correct significant

digits (sd), in the solutions,

graph{2+sin x (cos x-5sin x) [-10, `10. -.5, .5]}

graph{2+sin x (cos x-5sin x) [0.55 0.56 -3.14, 3.14]}

graph{2+sin x (cos x-5sin x) [2.5, 2.6 -3.14, 3.14]}

graph{2+sin x (cos x-5sin x) [3.1, 3.2 -3.14, 3.14]}

graph{2+sin x (cos x-5sin x) [5.7 5.71, -3.14, 3.14]}

graph{2+sin x (cos x-5sin x) [6.23, 6.26 -3.14, 3.14]}