How do you evaluate #cos(sin^-1(-1/4))# without a calculator?

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Answer 1 · 2016-08-29T07:09:43

#=sqrt 15/4#

Let #a=sin^(-1)(-1/4) in Q4#, wherein cosine is positive.

The given expression is

#cos a#

#=sqrt(1-sin^2 a)#

#=sqrt(1-(-1/4)^2)#

#=sqrt 15/4#.

Instead of the principal a or any other value in Q4, if we choose a

value in Q3, wherein cosine is also negative, the answer will be #-

sqrt 15/4#

For #sin a = -1/4#, the principal a in Q4 is -13.4775^o#, nearly.

An a in Q3 is #194.4775#, nearlr, and for this a,

#sin a =-1/4 and cos a =-sqrt 15/4#.

Yet, according to the convention for the choice of a as the principal

value, the sign is positive...

-