Step-by-step explanation:
imagine that point (-2, -5) on the coordinate grid.
in what quadrant is it ?
it is, where both x and y are negative : the bottom left (3rd) quadrant.
and this is the same quadrant for the circle and the placement of the angle of the point on the circumference to the x-axis (the basic diameter).
what signs do sine and cosine of angles in the third quadrant have ?
sine goes below the x-axis and is therefore negative.
cosine is to the left of the y-axis and is therefore negative.
that eliminates answer c (with positive sine and cosine values).
as tan(x) = sin(x)/cos(x)
and
cot(x) = cos(x)/sin(x)
both, tan and cot, must be positive (-/- = +).
that eliminates answers a and d.
but there is also a problem with the remaining answer b :
cot(x) = 1/tan(x), as the above definitions tell us.
but answer b says that they are both 5/2. and that cannot be.
so, there is no correct answer option.
therefore, let's calculate the real results, and then you need to find the right option in the original problem definition without typos.
the point again is (-2, -5). x = -2, y = -5.
the center of the circle is at the origin.
so, the radius of the circle is per Pythagoras
r = sqrt(x² + y²) = sqrt((-2)² + (-5)²) = sqrt(4 + 25) = sqrt(29)
sin(angle) = y/r = -5/sqrt(29) = -5×sqrt(29)/29
cos(angle) = x/r = -2/sqrt(29) = -2×sqrt(29)/29
tan = sin/cos = -5/sqrt(29) / -2/sqrt(29) = -5/-2 = 5/2
cot = cos/sin = -2/sqrt(29) / -5/sqrt(29) = -2/-5 = 2/5
csc(angle) = r/y = sqrt(29)/-5 = -sqrt(29)/5
sec(angle) = r/x = sqrt(29)/-2 = -sqrt(29)/2
again, except for the error in answer b about cot, it would all point to answer b.
