Given that tan0= -1, what is the value of sec0, for 3pi/2< 0< 2pi? a. -sqrt2 b. sqrt2 c. 0 d. 1 (the zeros in the question are theta)

Choice B. sqrt(2)

Draw out a right triangle in quadrant IV as you see in the attached image below. The horizontal and vertical legs are both 1 unit long. To ensure that the signs are properly set up, I am making the vertical leg BC have a label "-1" to mean this is below the x axis. Note how

tan(theta) = opposite/adjacent = BC/AB = -1/1 = -1

Use the pythagorean theorem to find that the hypotenuse AC is sqrt(2) units long

a^2 + b^2 = c^2

(1)^2 + (1)^2 = c^2

2 = c^2

c^2 = 2

c = sqrt(2)

The secant of theta is the ratio of the hypotenuse over the adjacent side, so we end up with

sec(theta) = hypotenuse/adjacent

sec(theta) = AC/AB

sec(theta) = sqrt(2)/1

sec(theta) = sqrt(2) which is why choice B is the answer