What is cos θ if sin θ = −8/10 and θ is in Quadrant III?

Drawing out the quadrants cos of an angle is negative in the third quadrant. Your drawing should indicate that x and y are negative. Then since we  are working with a right triangle, again your diagram should indicate that. Determine the value of the third side. You know one is - 6 and the other opposite the 90 degree angle, the hypotenuse is 10.   cos of an angle is adjacent divided by the hypotenuse.

I hope this helps if it did not help use this

cosine is negative in Quadrant III.

Use a pythagorean theorem to find the other leg.  a2 = c2 - b2

adjacent leg = √(102 - 82)

cos θ = -6/10