Simplify-cos-1-3-5-cos-x-4-5-sin-x-where-3pi-4-x-pi-4-

Question Number 20505 by Tinkutara last updated on 27/Aug/17

S i m p l i f y :

\cos^{- 1} (\frac{3}{5} \cos x + \frac{4}{5} \sin x), w h e r e

- \frac{3 π}{4} ⩽ x ⩽ \frac{π}{4}

Answered by ajfour last updated on 27/Aug/17

l e t \tan α = \frac{4}{3} \Rightarrow \sin α = \frac{4}{5},

a n d \cos α = \frac{3}{5}; α > x

θ = \cos^{- 1} [\cos α \cos x + \sin α \sin x]

= \cos^{- 1} [\cos (α - x)]

= α - x i f α - x ⩽ π

= 2 π - (α - x) i f α - x > π

T o s u m m a r i s e :

θ = {\begin{cases} \cos^{- 1} (\frac{3}{5}) - x; x ⩾ - π + \cos^{- 1} (\frac{3}{5}) \\ 2 π + x - \cos^{- 1} (\frac{3}{5}); x < - π + \cos^{- 1} (\frac{3}{5}) \end{cases}

Commented by Tinkutara last updated on 27/Aug/17

Thank you very much Sir!