Final answer:
To find cos a when tan a = 5/4 in quadrant I, we use the Pythagorean theorem to find the hypotenuse (√41) of the triangle. Then, the cosine function is adjacent over hypotenuse, giving us cos a = 4√41/41 in simplest radical form with a rational denominator.
Step-by-step explanation:
To find the exact value of cosine (cos a) given that the tangent of angle a is 5/4 and the angle is in quadrant I, we can use the Pythagorean identity which relates sine, cosine, and tangent. Since tan a = opposite/adjacent (opposite side over the adjacent side to angle a in a right-angled triangle), and tangent a = 5/4, we can determine the sides of the triangle. Recall that tan² a + 1 = sec² a, which rearranges to 1 = cos² a + sin² a. Applying the Pythagorean theorem to the right triangle, we get (adjacent side)² + (opposite side)² = (hypotenuse)². Thus, with an opposite side of 5 and an adjacent side of 4, the hypotenuse can be calculated as h = √(5² + 4²) = √(25 + 16) = √41. The cosine function is given by the adjacent side over the hypotenuse, so cos a = 4/√41. To express this with a rational denominator, we multiply the numerator and denominator by √41 to get cos a = 4√41/41, which is the final answer in simplest radical form with a rational denominator.