Final answer:
The exact value of sec(5π/4) is -√2, derived by taking the reciprocal of cos(5π/4) which is -√2/2.
Step-by-step explanation:
To find the exact value of sec(5π/4), we must recall that the secant function is the reciprocal of the cosine function. Therefore, sec(θ) = 1 / cos(θ). The angle 5π/4 is located in the third quadrant of the unit circle, where the cosine value is negative. Specifically, the coordinates of the point on the unit circle at an angle of 5π/4 are (-√2/2, -√2/2), where the x-coordinate represents the cosine value.
Thus, cos(5π/4) is -√2/2. Taking the reciprocal of this value so that we can find sec(5π/4), we have:
sec(5π/4) = 1 / cos(5π/4)
sec(5π/4) = 1 / (-√2/2)
sec(5π/4) = -2 / √2
sec(5π/4) = -√2
This is the exact value of the secant function at the given real number.