Sure, let's analyze the expression [tex]\(\cos^{-1}\left[\frac{3}{5} \cos x + \frac{4}{5} \sin x\right]\)[/tex].
Firstly, observe that the given expression inside the arccos function is a linear combination of cosine and sine. This can be rewritten in a more convenient form using trigonometric identities and phase shifts.
Consider the expression:
[tex]\[ y = \frac{3}{5} \cos x + \frac{4}{5} \sin x \][/tex]
We can rewrite this as:
[tex]\[ y = A \cos x + B \sin x \][/tex]
where [tex]\(A = \frac{3}{5}\)[/tex] and [tex]\(B = \frac{4}{5}\)[/tex].
To simplify this expression, we can use the identity for a phase-shifted cosine function. The identity states:
[tex]\[ A \cos x + B \sin x = R \cos(x - \phi) \][/tex]
where:
[tex]\[ R = \sqrt{A^2 + B^2} \][/tex]
and
[tex]\[ \phi = \tan^{-1}\left(\frac{B}{A}\right) \][/tex]
Let's find [tex]\(R\)[/tex]:
[tex]\[ R = \sqrt{\left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2} = \sqrt{\frac{9}{25} + \frac{16}{25}} = \sqrt{1} = 1 \][/tex]
So, we have [tex]\( R = 1 \)[/tex].
Next, we determine the phase shift [tex]\(\phi\)[/tex]:
[tex]\[ \phi = \tan^{-1}\left(\frac{4}{5} \div \frac{3}{5}\right) = \tan^{-1}\left(\frac{4}{3}\right) \][/tex]
The value of [tex]\(\tan^{-1}\left(\frac{4}{3}\right)\)[/tex] is a constant, and it simplifies to approximately:
[tex]\[ \phi = 0.9272952180016123 \][/tex]
Given this, the initial expression [tex]\( y = \frac{3}{5} \cos x + \frac{4}{5} \sin x \)[/tex] can be rewritten as:
[tex]\[ y = \cos(x - \phi) \][/tex]
Now, we need to find [tex]\(\cos^{-1}(y)\)[/tex]:
[tex]\[ \cos^{-1}\left[\frac{3}{5} \cos x + \frac{4}{5} \sin x\right] = \cos^{-1}(\cos(x - \phi)) \][/tex]
Since [tex]\(\cos^{-1}(\cos(\theta)) = \theta\)[/tex] for [tex]\(\theta\)[/tex] in the range [tex]\([0, \pi]\)[/tex], we get:
[tex]\[ \cos^{-1}(\cos(x - \phi)) = |x - \phi| \][/tex]
because the arccos function will give us an angle in the principal value range [tex]\([0, \pi]\)[/tex].
Thus, the final value we seek is:
[tex]\[ \cos^{-1}\left[\frac{3}{5} \cos x + \frac{4}{5} \sin x\right] = x - 0.9272952180016123 \][/tex]
This completes the detailed step-by-step solution for the given question.
