they are shown below:
the graph shows that you have intersection points at:
x = 1.008
x = 2.133
x = 4.381
x = 5.043
to solve this algebraically, you have to do the following:
start with:
5cos(2x)+2=sin(x)-1
subtract sin(x) from both sides of the equation and add 1 to both sides of the equation to get:
5cos(2x) + 2 - sin(x) + 1 = 0
combine like terms to get:
5cos(2x) + 3 - sin(x) = 0
cos(2x) is equal to cos^2(x) - sin^2(x) which is equal to 1 - sin^2(x) - sin^2(x) which is equal to 1 - 2sin^(x).
replace cos(2x) with 1 - 2 sin^2(x) and you get:
5 * (1 - 2sin^2(x)) + 3 - sin(x) = 0
simplify to get:
5 - 10sin^2(x) + 3 - sin(x) = 0
combine like terms to get:
8 - 10sin^2(x) - sin(x) = 0
rearrange the terms to get:
-10sin^2(x) - sin(x) + 8 = 0
multiply both sides of this equation by -1 to get:
10sin^2(x) + sin(x) - 8 = 0
let y = sin(x) to get:
10y^2 + y - 8 = 0
solve this quadratic equation using the quadratic formula to get:
y1 = .8458236434
y2 = -.9458236434
since you have set y = sin(x), then you can replace y1 with sin(x1) and you can replace y2 with sin(x2) to get:
sin(x1) = .8458236434
sin(x2) = -.9458236434
solve for x1 and x2 to get:
x1 = arcsin(.8458236434) = 1.00810724
x2 = arcsin(-.9458236434) = -1.240122079
sine is positive in quadrants 1 and 2.
x1 = 1.00810724 is in quadrant 1.
the equivalent angle in quadrant 2 is equal to pi - 1.00810724 which is equal to 2.133485414.
so you have:
x1a = 1.00810724 radians
x1b = 2.133485414 radians.
sine is negative in quadrants 3 and 4.
x2 = -1.240122079 is in quadrant 4.
the equivalent angle in quadrant 1 is equal to - (-1.240122079) which is equal to 1.240122079.
1.240122079 is the equivalent angle in quadrant 1.
now that you found the equivalent angle in quadrant 1, you can use the standard formula to find the equivalent angle in quadrant 3 and quadrant 4.
the equivalent angle in quadrant 3 is equal to pi + 1.240122079 which is equal to 4.381714733.
the equivalent angle in quadrant 4 is equal to 2*pi - 1.240122079 which is equal to 5.043063228.
you now have:
x2a = 4.381714733.
x2b = 5.043063228.
your solutions are:
x1a = 1.00810724 radians = 1.008 rounded to 3 decimal places.
x1b = 2.133485414 radians = 2.133 rounded to 3 decimal places.
x2a = 4.381714733 radians = 4.382 rounded to 3 decimal places.
x2b = 5.043063228 radians = 5.043 rounded to 3 decimal places.
the solutions shown in the graph confirm these values.