Find sec(theta) if tan(theta)=sqrt(3)/3, and sin(theta)<0.
you were given $\displaystyle \tan{t} = \frac{\sqrt{3}}{3}$ ... a (+) value
you were also told $\displaystyle \sin{\theta} < 0$ ... a (-) value
those two pieces of information tell you what quadrant $\displaystyle \theta$ is in ... so you can determine the correct sign for $\displaystyle \sec{\theta}$
Find sec(theta) if tan(theta)=sqrt(3)/3, and sin(theta)<0.
tan is + and sin is < 0, that is - ; that is in the 3rd quadrant.
But in the 3rd quadrant only tan and cot are positive, the rest negative.
, from this we have,
sec = -(2/3)(sqrt 3)
the negative sign ( - ) is introduced since it is in the 3rd quadtant