use-right-triangles-to-explain-why-cos-1-x-sin-1-x-pi-2-

Question Number 126701 by Eric002 last updated on 23/Dec/20

u s e r i g h t t r i a n g l e s t o e x p l a i n

w h y {c o s}^{- 1} (x) + {s i n}^{- 1} (x) = π / 2

Answered by ebi last updated on 23/Dec/20

l e t {s i n}^{- 1} (x) = θ, t h e n x = s i n θ

s i n θ = c o s (\frac{π}{2} - θ)

{c o s}^{- 1} (x) = \frac{π}{2} - θ

{c o s}^{- 1} (x) = \frac{π}{2} - {s i n}^{- 1} (x)

{c o s}^{- 1} (x) + {s i n}^{- 1} (x) = \frac{π}{2}

Answered by mathmax by abdo last updated on 24/Dec/20

let f (x) = arcosx + arcsinx - \frac{π}{2}

\forall x \in [- 1, 1] f^{^{'}} (x) = - \frac{1}{\sqrt{1 - x^{2}}} + \frac{1}{\sqrt{1 - x^{2}}} = 0 \Rightarrow f (x) = c

f (0) = \frac{π}{2} - \frac{π}{2} = 0 \Rightarrow \forall x \in [- 1, 1] f (x) = 0 \Rightarrow arcosx + arcsinx = \frac{π}{2}