Question

If tan\alpha =-(2)/(5) and \alpha is in Quadrant IV, find cos\alpha .

Answer 1

To find cosα when tanα = -2/5 and α is in the fourth quadrant, we use the Pythagorean identity to solve for secα, and then find cosα by taking the reciprocal, which is approximately 0.927 after rationalizing.

Step-by-step explanation:

If tanα = -(2)/(5) and α is in Quadrant IV, to find cosα, we can use the Pythagorean identity associated with tangent and cosine, which is tan2(α) + 1 = sec2(α), where secα = 1/cosα. Since we know that tanα = -(2)/(5), we can solve for secα and subsequently for cosα.

First, we calculate sec2(α) as follows:

• tan2(α) = (2/5)2 = 4/25,
• sec2(α) = tan2(α) + 1 = 4/25 + 1 = 29/25,
• secα = ±sqrt(29/25) = ±sqrt(29)/5.

Because α is in the fourth quadrant, where cosine is positive, we choose the positive value for secα, so secα = sqrt(29)/5 and therefore cosα = 5/sqrt(29) or cosα ≈ 0.927 after rationalizing the denominator.