How to fix a reduce function error when using trigonometric functions?

Question

For example, if I'm trying to find out when the function is greater than or equal to 20 it gives an error. I also trying defining my trig equation as a function and tried to solve that (e.g Reduce[h[t]>= 20]) but that didn't work either. Any suggestions?

Answer 1

Specify a range for t to make Reduce work:

Reduce[{19 - 17 Cos[Pi t/8] >= 20, 0 < t < 16}, t]
(*(8 ArcCos[-(1/17)])/\[Pi] <= t <= (16 \[Pi] - 8 ArcCos[-(1/17)])/\[Pi]*)  

Answer 2

First plot what you want to calculate

Plot[{19 - 17 Cos[(Pi* t)/8], 20}, {t, -20, 20}, Filling -> {2 -> Top}]

then select the area of interest, e.g.

Reduce[{19 - 17 Cos[(Pi* t)/8] >= 20, 0 < t < 4 Pi}, t] // N
4.14988 <= t <= 11.8501

Answer 3

Clear["Global`*"]

Restrict the domain to Reals

sol = Reduce[19 - 17 Cos[(Pi*t)/8] >= 20, t, Reals] /. C[1] -> n

(* n ∈ Integers && (16 n π + 8 ArcCos[-(1/17)])/π <= t <= (
  16 π + 16 n π - 8 ArcCos[-(1/17)])/π *)

Table[{n, sol // N}, {n, -2, 2}] // Grid[#, Frame -> All] &