Final answer:
To find the exact value of cosine, we use the relationship between cosine and tangent and the Pythagorean identity. Given that tan(a) = 5/4 and angle a is in Quadrant I, the exact value of cosine(a) is 5/4.
Step-by-step explanation:
To find the exact value of cosine (a), we can use the relationship between cosine and tangent. Since tan(a) = 5/4, we know that tangent is positive in Quadrant I. In Quadrant I, cosine is positive and sine is also positive. So, we can use the Pythagorean identity to find the value of cosine.
First, let's find the value of sin(a) using the Pythagorean identity: sin(a) = sqrt(1 - cos^2(a)). We can substitute the given value of tan(a) into this equation to get: sin(a) = sqrt(1 - (5/4)^2) = sqrt(1 - 25/16) = sqrt(16 - 25)/4 = sqrt(-9)/4 = (i*3)/4.
Since we know that cosine is positive in Quadrant I, we can use the relationship between cosine and sine: cos(a) = sqrt(1 - sin^2(a)). Substitute the value of sin(a) into this equation: cos(a) = sqrt(1 - [(i*3)/4]^2) = sqrt(1 - (i*3)^2/16) = sqrt(1 - (-9)/16) = sqrt(16 + 9)/4 = sqrt(25)/4 = 5/4.