cosecant is equal to 1/sine, which is positive in quadrant 1 and 2, and negative in quadrant 3 and 4.
if the cosecant, which is equal to 1/sine, is negative and the cotangent, which is equal to 1/tangent, is positive, you have to be in quadrant 3.
you are given that cosecant theta = -2.
this means that 1/sine theta = -2.
solve for sine theta to get:
sine theta = -1/2.
in the first quadrant, sine theta would be equal to 1/2 and the angle would be equal to 30 degrees.
the same angle in the third quadrant is equal to 180 + 30 degrees = 210 degrees.
the sine of 210 degrees is equal to -1/2.
the cosecant of 210 degrees is equal to 1/sine 210 degrees = 1/(-1/2) = -2.
the cosine of 210 degrees is equal to -sqrt(3)/2.
the secant of 210 degrees is equal to 1/cosine 210 degrees = 1/(-sqrt(3)/2) = 2/-sqrt(3) = -2*sqrt(3)/3.
the tangent of 210 degrees is equal to -1/-sqrt(3) = 1/sqrt(3) = sqrt(3)/3.
the cotangent of 210 degrees is equal to 1/(sqrt(3)/3) = 3/sqrt(3) = 3*sqrt(3)/3 = sqrt(3).
the following diagram shows you what the angle looks like in quadrant 3 where it resides.
the reference angle is the equivalent angle in the first quadrant.
it has the same trigonometric function values except for the sign.
in the first quadrant all the trigonometric values are positive.
in the third quadrant, the trigonometric functions have the same values, excpt:
sine and cosecant are negative.
cosine and secant are negative.
tangent and cotangent are positive.
here are the trigonometric values of the special angles, 30 degrees and 210 degrees being among them.
https://www.mcckc.edu/tutoring/docs/br/math/calc_trig/Special_Angles_Chart.pdf
30 degrees and 210 degrees are highlighted below.
