What is the value of cosθ, given that tanθ=8/3 and cosθ< 0? simplify radicals and rationalize the denominator, if necessary.

1. Check the picture below.

2. Tan∅=8/3 so let's construct the right triangle ABC with BC=8, BA=3, m(B)=90° and m(A) =∅. 

3. By Pythagorean theorem, AC=

4. cos∅=adjecent side/hypothenuse=

5. Cos∅ is negative so ∅ is in an angle in the second or third quadrant. The assumption that ∅ is and acute angle of a right triangle only makes the solution more practical. The only thing that changes, is sign, which depends on the quadrant.

6. As cos ∅ is negative, cos∅=- .