Pre calculus HW

Question

Find the exact value of the given expression.

sec(2 sin^-1(1/8))

Answer 1

Let θ = Sin^-1 (1/8). Then sinθ = 1/8.

So, sec(2θ) = 1 / cos(2θ) = 1 / [1 - 2sin²θ] = 1 / [1 - 1/32] = 1 / (31/32) = 32/31

Answer 2

32/31 = 1 1/31 = the "exact" answer

sin^-1(1/8) = about 7.18 degrees

(yes, I know you want the "exact" value, but check whatever you get with a close approximation, to see if you're even in the right range of answers)

2sin^-1(1/8) = about 2(7.18) = 14.36

sec14.36 = 1/cos14.36 = about 1.03

whatever you get as an exact answer should be close to 1.03

secants are all either >1 or <-1, so 1.03 is in the feasible range

sec(2sin^-1(1/8)

exact answer is sec(2sin^-1(1/8) = 32/31 = about 1.032258 = about 1.03

sinA = 1/8 where sin^-1(1/8) = A

let B =2A

sin B = sin2A = 2sinAcosA = 2(1/8)(sqr63/80 = sqr63/32

B = sin^-1(sqr63/32)

secB = 32/31 = hypotenuse over adjacent side

adjacent side squared = 32^2 -63 = 1024-63 = 961

sqr961 = 31

secB = 32/31 exactly which is very close to 1.03