If sec theta = 5/4, find csc(theta+pi/3)

Question

I don't know what to do to even start this problem. 

Answer 1

let theta=x

secx=5/4

secx=1/cosx

1/cosx=5/4

cosx=4/5

in the right triangle if cosx=4/5 and cos=adj/hyp, then sin=opp/hyp and sinx=3/5 because you have a 3-4-5 right triangle

cscx=1/sinx

csc(x+60)=1/sin(x+60)   (pi/3=180/3=60)

1/sin(x+60)=

1/[(sinxcos60)+(cosxsin60)]=

1/[(3/5*1/2)+(4/5*√3/2)]=

1/[(3/10)+(4√3/10)]=

1/([3+4√3]/10)=

10/(3+4√3)

rationalize the denominator

10(3-4√3)/(3+4√3)(3-4√3)

10(3-4√3)/(9-16*3)

10(3-4√3)/(9-48)

10(3-4√3)/(-39)