If csc theta=13/5 and cos theta <0, then cot theta equals

Question

If csc theta = 13/5 and cos theta <0, then cot theta equals

Answer 1

The cscθ=1/sinθ. Therefore, 13/5=1/sinθ or sinθ=5/13. Sine is the opposite side divided by the hypotenuse, or sin=opp/hyp. Thus, the opposite side = 5 and the hypotenuse = 13. Knowing that opp.²+adj.²=hyp.² we can find out what the adjacent (adj.) side equals:  5²+adj.²=13²

                                               25+adj.²=169

                                               adj.²=169-25

                                               adj.²=144

                                               adj.=±12

Cosine = adj./hyp. Because cosθ<0, cosθ=-12/13.

Knowing that cot=1/tan and tan=sin/cos, this makes cot=cos/sin. Therefore, we have cotθ=(-12/13)/(5/13)=-12/5.