Question 1117358
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You are given  sin(t) = {{{-4/9}}}  and "t" is in the 4-th quadrant.


In QIV cos(t) is positive, and therefore


    cos(t) = {{{sqrt(1-sin^2(t))}}} = {{{sqrt(1-(-4/9)^2)}}} = {{{sqrt(1 - 16/81)}}} = {{{sqrt((81-16)/81)}}} = {{{sqrt(65/81)}}} = {{{sqrt(65)/9}}}.


Then tan(t) = {{{sin(t)/cos(t)}}} = {{{((-4/9))/((sqrt(65)/9))}}} = {{{-4/sqrt(65)}}} = rationalize the denominator = {{{(-4*sqrt(65))/65}}}.


Next  sec(t) = {{{1/cos(t)}}} = {{{1/((sqrt(65)/9))}}} = {{{9/sqrt(65)}}} = rationalize the denominator = {{{(9*sqrt(65))/65}}}.
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