Q: Let X be an angle in quadrant III such that cos(X) = -12/13. Find the exact values of csc(X) and cot(X).
This will be a little hard to explain, since I cannot currently provide a picture, but it should be manageable! Draw a picture on your own paper and follow along with me.
Draw a right triangle and label one of the angles X (just not the right angle). We are going to ignore the negative sign for now and then take it in to account later! Since the cos(X) = 12/13, the side adjacent to X is 12 and the hypotenuse is 13. Use the Pythagorean Theorem to find the missing side (the side opposite X). You will find that the opposite side is 5.
So, Adj = 12, Opp = 5 and Hyp = 13
Now, let’s consider the negative sign: Since X is a QIII angle, we know that only tangent and cotangent are positive values in that quadrant — all other trig functions in QIII are negative.
Now, we need to find the csc(X). “csc” is the ratio of the hyp / opp [and, it will be negative]
So, csc(X) = -13/5
And, cot(x) is the ratio of the adj / opp [and, it will be positive]. So, cot(X) = 12/5