Given value (without calculator): sec[sin^-1(-1/5)] Give value?

Give the value

1 Answer
Apr 26, 2018

# (5sqrt6)/12#.

Explanation:

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

#sin^-1x=theta; |x| le 1 hArr sintheta=x, |theta| le pi/2#.

Since #|-1/5| le 1, sin^-1(-1/5)" is defined"#.

Let, #sin^-1(-1/5)=theta, theta in [-pi/2,pi/2]=Q_4uuQ_1#.

Then, #sintheta=-1/5 lt 0. :. theta in Q_4#.

Now, with #sin^-1(-1/5)=theta#,

#"The reqd. value"=sectheta............(diamond)#.

Thus, we know the value of #sintheta# and require that of

#sectheta=1/costheta#.

Now, #cos^2theta=1-sin^2theta=1-(-1/5)^2=1-1/25=24/25#.

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

Knowing that #theta in Q_4, costheta=+(2sqrt6)/5#.

#(diamond) rArr" the reqd. value"=5/(2sqrt6)=(5sqrt6)/12#.