Question 754320
You can start by drawing a right triangle in the third quadrant. They tell you that the cosecant of theta is -6/5. The cosecant is the ratio of the length of the hypotenuse to the length of the side opposite theta.

The hypotenuse always has a positive value, and you would label the hypotenuse "6" and the side opposite the angle "-5". You will then also want to find the adjacent side of your triangle (the side lying on the x-axis). Use the Pythagorean theorem to find this side.



You know a^2 + b^2 = c^2. For your triangle, then, (-5)^2 + b^2 = 6^2



25 + b^2 = 36 --> b^2 = 11 --> b = sqrt(11). And, since your triangle is in the 3rd quadrant, the adjacent side of your triangle will have a negative sign, so b = -sqrt(11)


So you have:

hypotenuse = 6

opposite = -5

adjacent = -sqrt(11)


You can use these values to find the tangent of theta and the cosine of theta.



Tan (theta) = opposite/adjacent = -5/-sqrt(11)


Since this has a radical expression in the denominator, they may want you to "rationalize" the denominator. To do that, you could multiply by sqrt(11)/sqrt(11):


(-5/-sqrt(11)) X (sqrt(11)/sqrt(11)) = 5sqrt(11)/11     This should be the tangent of your angle.





Cos (theta) = adjacent/hypotenuse  = -sqrt(11)/6    This should be the cosine of your angle.



I hope this was helpful!