How do you find the exact 6 trigonometric ratios for the angle x whose radian measure is ##(5pi)/6##?

You should have them memorized from the unit circle.
The x-coordinate is the cosine value for that angle.
The y-coordinate is the sine value for that angle.
Tangent is ##sin / cos##, or in this case, ##y / x##.
Secant is the reciprocal of cosine.
Cosecant is the reciprocal of sine.
Cotangent is the reciprocal of tangent (or ##x / y##)
Memorization tip: Memorize only the first quadrant, then just change the positive/negative of the coordinate for other quadrants!