Question 288206
Remember that {{{tan(x)=sin(x)/cos(x)}}}. So if {{{tan(a)>0}}}, then {{{sin(a)/cos(a)>0}}}. Now because we already know that {{{cos(a)>0}}}, we can multiply both sides of {{{sin(a)/cos(a)>0}}} by {{{cos(a)}}} to get {{{sin(a)>0}}} (and we don't have to worry about if the sign will flip or not).



So if {{{tan(a)>0}}} and {{{cos(a)>0}}}, then {{{sin(a)>0}}} as well. Recall that the cosine of a given angle will produce the 'x' value of the terminal side on the unit circle. So if {{{cos(a)>0}}}, then we can say that {{{x>0}}} for the given point (x,y) which is the endpoint of the terminal side.



Similarly if {{{sin(a)>0}}}, then {{{y>0}}} (since sine is associated with the 'y' value of the point on the unit circle). This means that both 'x' and 'y' are positive which places the terminal side in the first quadrant.