cos.

cos(theta)=-1/2 and tan(theta)=sq3 find csc

 

Cos is neg and tan is pos so theta is in the 3rd quad.

Now put this aside for now and look for the 1st quad equivalent angle.

 

Draw a right angled triangle and call one of the acute angles theta.

The adjacent side is 1, the hypotenuse is 2, Using pythagoras's theorum you can determing that the third side is sqrt(3)

 

so tan (theta)=sqrt(3)

cos(theta)=1/2

cosec(theta)=1/sin2(theta)=hyp/opp = 2/sqrt(3) = (2sqrt3)/3

But theta it is in the 3rd quad so  sine and cosec are negative

 

cosec(theta)= - (2sqrt3)/3             (as an aside, theta is 240 degrees)

Cos -1/2=.9999619, tan(3)^2=tan 9=.15838444,

But tan=sine/cosine, so we have

.15838444=sine/.9999619

Sine=.15838444 X .9999619=.15837840,

Csc=1/sine=1/.15837740=6.3139920. That's it, if by tan=sq3 you mean 3^2. If not, then the answer will be different. Good luck.