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

Aug 29, 2016

$= \frac{\sqrt{15}}{4}$

#### Explanation:

Let $a = {\sin}^{- 1} \left(- \frac{1}{4}\right) \in Q 4$, wherein cosine is positive.

The given expression is

$\cos a$

$= \sqrt{1 - {\sin}^{2} a}$

$= \sqrt{1 - {\left(- \frac{1}{4}\right)}^{2}}$

$= \frac{\sqrt{15}}{4}$.

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 $\sin a = - \frac{1}{4}$, the principal a in Q4 is -13.4775^o#, nearly.

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

$\sin a = - \frac{1}{4} \mathmr{and} \cos a = - \frac{\sqrt{15}}{4}$.

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

value, the sign is positive...

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