Finding Solutions

Question

Find the solutions of the equation that are in the interval [0, 2π)

sec5 θ = 4 sec θ

Answer 1

sec^5(x) = 4secx

sec^4(x) = 4

1/cos^4(x) = 4

cos^4(x) = 1/4=.25

cosx = 4th root of 1/4

cos^2(x) =square root of 1/4 = 1/2

cosx = sqr(1/2) = 1/sqr2 = sqr2/2

x = 45 degrees = pi/4 or 7pi/4 = 315 degrees

x = pi/4, 7pi/4

also secx = 0

but that's impossible as |secx| always >1, unless you want some "imaginary" or other non-real surreal solution

so only solutions are pi/4 and 7pi/4 or 45 and 315 degrees