Sec(sin^-1(-1/5)) Give value?

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It says not to use a calculator so I'm just not sure how to find it without

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Answer 1 · 2017-11-20T05:05:27

# 5/(2sqrt6).#

Recall the Definition of the #sin^-1" function :"#

#sin^-1x=theta, -1 le x le 1 iff sintheta=x, -pi/2 le theta le pi/2.#

So, let #sin^-1(-1/5)=theta rArr sin theta=-1/5, -pi/2 le theta le pi/2.#

Note that #sintheta lt 0 rArr theta in [-pi/2,0].#

Now, #"the Reqd. Value="sec(sin^-1(-1/5))=sectheta=1/costheta,#

#"But, "sintheta--1/5 rArr cos^2theta=1-sin^2theta=1-(-1/5)^2,#

#=1-1/25=24/25.#

#:. costheta=pmsqrt(24/25)=+-(2sqrt6)/5.#

Remembering that, #theta in [-pi/2,0]=Q_(IV),# we have,

#costheta=+(2sqrt6)/5.#

# rArr sec(sin^-1(-1/5))=sectheta=5/(2sqrt6).#