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Explanation:
Draw a right triangle in the first quadrant, aka northeast quadrant, as shown in the diagram below.
Cosecant involves the ratio of hypotenuse over opposite. It's the reciprocal of sine
sin(angle) = opposite/hypotenuse
csc(angle) = hypotenuse/opposite
since csc = 1/sin
Use the pythagorean theorem to solve for b in the diagram
a^2+b^2 = c^2
16^2+b^2 = 65^2
256+b^2 = 4225
b^2 = 4225-256
b^2 = 3969
b = sqrt(3969)
b = 63
So this is the adjacent side. The secant ratio involves hypotenuse over adjacent (it's the reciprocal of cosine).
sec(angle) = hypotenuse/adjacent
sec(A) = 65/63
