Given csc(A) = 60/16 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!

Question

asked Mar 14, 2022 218k views

1 Answer

Cosmin Vasii · Answer 1 · 2022-03-20T13:53:04+0000

Answer:

(15 √(209) )/(209)

Explanation:

Objective: Understand and work with trig identies.

Recall multiple trig identies and manipulate them to get from cosecant to secant.

Given

$\csc(a) = (60)/(16)$

Apply reciprocal identity csc a = 1/sin a.

$\sin(a) = (16)/(60)$

Apply pythagorean identity to find cos a.

$( (16)/(60)) {}^(2) + \cos(x) {}^(2) = 1$

$(256)/(3600) + \cos(x) {}^(2) = 1$

$\cos(x) {}^(2) = (3600)/(3600) - (256)/(3600)$

We can simplify both expression

$\cos(x) {}^(2) = (225)/(225) - (16)/(225)$

$\cos(x) = ( √(209) )/(15)$

Cosine is positve on quadrant 1 so that cos(a)

Apply reciprocal identity sec a= 1/ cos a.

The answer is

$\sec(a) = (15)/( √(209) )$

Rationalize the denominator.

(15)/( √(209) ) * ( √(209) )/( √(209) ) = (15 √(209) )/(209)