Question 169169
Let {{{x=sec^(-1)(-2)}}}. So this means that {{{sec(x)=-2}}} which further tells 



{{{sec(x)=-2}}} Start with the given equation



{{{1/cos(x)=-2}}} Replace secant with {{{1/cosine}}}. Take note that cosine is negative (since 1 divided by a negative number is negative). This means that the angle MUST be in either the 2nd or 3rd quadrants.



{{{1=-2cos(x)}}} Multiply both sides by {{{cos(x)}}}



{{{-1/2=cos(x)}}} Divide both sides by -2



{{{arccos(-1/2)=x}}} Take the arccosine (or inverse cosine) of both sides to isolate x




{{{x=-pi/3}}} or {{{x=2pi/3}}} Take the arccosine of {{{-1/2}}} to get {{{-pi/3}}} or {{{2pi/3}}}



Since it was stated above that the angle had to be in quadrants 2 or 3, this means that we can ignore {{{x=-pi/3}}}




So {{{x=2pi/3}}} which means that {{{sec^(-1)(-2)=2pi/3}}}