Question

Given csc(A) = 65/16 and that angle A is in Quadrant I, find the exact value of sec A in simplest radical form using a rational denominator . Someone please help me!

Answer 1

Answer:  65/63

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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