Can someone please help. Let theta be an angle in quadrant III such that sin(theta)=-7/8. Find the exact values of sec and cot.?

y = -7, r = 8
Pythag. x = -sqrt (64 - 49) = - sqrt (15)

sec T = r/x = 8 / [ - sqrt (15) ] << rationalize / simplify

cot T = x/y = [ - sqrt (15) ] / -7 <<< simplify

the reference triangle for Θ is { - √15 , - 7 , 8 } = { adj , opp , hyp }

Since sine is opposite divided by hypotenuse, you can imagine that theta is an angle of a right triangle with one side 7 units long and the hypotenuse 8 units long. By Pythagorean Theorem, we have that the remaining leg is

sqrt(64-49) = sqrt(15) units long.

secant is hypotenuse over adjacent. And since theta is in the third quadrant, secant is negative. So,

sec(theta) = -8/sqrt(15).

cotangent is adjacent over opposite, and it is positive in the third quadrant. So,

cot(theta) = sqrt(15)/7