Final answer:
To find the exact value of sec(theta), we can use the given information about the tangent and cosecant of the angle. By using trigonometric identities, we can determine the sine and cosine of theta and then find the secant. The exact value of sec(theta) is 3/4.
Step-by-step explanation:
To solve this problem, we can use the given information about the tangent and cosecant of the angle to find the exact value of the secant. We know that tan(theta) = -5/3 and csc(theta) > 0. First, we can determine the sine of theta using the given information. Since csc(theta) is the reciprocal of sin(theta), we know that sin(theta) = 1/csc(theta). Therefore, sin(theta) = 1/csc(theta) = 1/(1/sin(theta)) = sin(theta).
Next, we can use the Pythagorean identity, which states that sin^2(theta) + cos^2(theta) = 1, to find the cosine of theta. Since we know the sine of theta is -5/3, we can square it to get sin^2(theta) = 25/9. Plugging this into the Pythagorean identity and solving for cos^2(theta), we get cos^2(theta) = 1 - sin^2(theta) = 1 - 25/9 = 16/9. Taking the square root of both sides gives us cos(theta) = sqrt(16/9) = 4/3.
Finally, we can find the secant of theta using the identity sec(theta) = 1/cos(theta). Plugging in the value we found for cos(theta), we get sec(theta) = 1/(4/3) = 3/4.