pre calc question

Question

Write  in terms of  if  is in quadrant II: 

, 

Answer 1

sect = A/sqr(B+C^2)

sect=1/cost, tant =sint/cost= sintsect

cost = sqr(B+C^2)/A = adjacent side/hypotenuse

sint = opposite side/hypotenuse

opposite side^2 = A^2 - (sqr(B+C^2)^2=A^2-B-C^2

sint =sqr(A^2-B-C^2)/A

tant = (sqr(A^2-B-C^2)/A)/(A/sqr(B+C^2))

tant =sqr((A^2-B-C^2)/sqr(B+C^2))

tant = -sqr((A^2-B-C^2)/(B+C^2))

or an alternative method:

tant = -sqr(sec^2(t) -1)

= -sqr(A^2/(B+C^2) -1)= sqr((A^2 -B-C^2)/(B+C^2)

in quadrant II, tant <0 so just take the negative square root