Question

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

Answer 1

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=
`\sqrt{ 8^(2) + 3^(2) } = √(64+9) = √(73)`

4. cos∅=adjecent side/hypothenuse=
`( 3)/( √(73))= ( 3)/( √(73)) (√(73))/(√(73))= (3√(73))/(73)`

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∅=-
`(3√(73))/(73)` .