Question

solve the equation for x, where x is restricted to the given interval.
y = cos(x + 5), for x in -5,π - 5
x =

Explanation:

Step1: Apply inverse cosine

Given \( y = \cos(x + 5) \), we can write \( x+5=\cos^{-1}(y) \) (since for \( t=x + 5\), when \( t\in[0,\pi]\) (because \(x\in[-5,\pi - 5]\), then \(x + 5\in[0,\pi]\)), \(y=\cos t\) and \(t = \cos^{-1}(y)\)).

Step2: Solve for \(x\)

Subtract 5 from both sides of the equation \(x+5=\cos^{-1}(y)\). We get \(x=\cos^{-1}(y)-5\).

Answer:

\(x=\cos^{-1}(y)-5\)