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

1 Answer
Jan 20, 2017

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]}