How do you simplify ${cos}^{- 1} [sin (- \frac{π}{4})]$ $cos^-1 [sin (-pi/4) ]$ ?

Question

8604 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-07-11T02:13:57

Use $cos ({cos}^{- 1} a) = a$ $cos (cos^(-1)a)=a$ .

With the principal values as the angles,

$cos (\frac{3 π}{4}) = sin (- \frac{π}{4}) = - \frac{1}{\sqrt{2}}$ $cos ((3pi)/4)=sin (-pi/4)=-1/sqrt 2$

Now, the given expression = ${cos}^{- 1} (cos (\frac{3 π}{4})) = \frac{3 π}{4}$ $cos^(-1)(cos ((3pi)/4))= (3pi)/4$ .

Of course, the general value is

$a=2npi+-(3pi)/4, n=0, +-1, +-2, +-3, ...$ .

This sorting comes from

$cos^(-1)cos$ (general value); by choice.

How do you simplify cos^-1 [sin (-pi/4) ]cos−1[sin(−π4)]cos^-1 [sin (-pi/4) ]?