91Ó°ÊÓ

Recall that a cosine function has the general form \(f(x) = A \cos (Bx - C)\), where A is the amplitude, B is the horizontal stretch factor, and C is the phase shift of the function. In our case, the given function is \(a \cos (bx - 4)\), which can be written as \(a \cos (b(x - \tfrac{4}{b}))\). Here, the amplitude is a, the horizontal stretch factor is b, and the phase shift is \(\tfrac{4}{b}\), which is a shift to the right.

The cosine function has the following identity: \[\cos(-x) = \cos(x)\] Now, recall that the given function is \(a \cos (bx - 4)\). Using the identity, we can rewrite the function as: \(a \cos (-(4 - bx))\)

Now, we need to find a positive constant \(\tilde{c}\) that, when added to the argument, gives us the same function as before. Using the identity in Step 2, we have, \(a \cos (-(4 - bx)) = a \cos(bx - 4)\) From our analysis of the cosine function, we know that the period of the function is \(2\pi\). Therefore, if we can find a positive constant \(\tilde{c}\) such that \(4 - \tilde{c}\) is greater than or equal to 0 and divisible by \(2\pi\), the given function can be rewritten in the required form. Let's assume \(\tilde{c} = 2k \pi + 4\), where \(k\) is an integer. Now, we can rewrite the given function with \(\tilde{c}\) as follows: \(a \cos (bx + 2k \pi + 4 - 4) = a \cos (bx + 2k \pi)\) Since \(\cos (bx + 2k \pi) = \cos (bx)\), we have achieved the desired form of the given function: \(a \cos (bx + \tilde{c}) = a \cos (bx)\)

Given a function of the form \(a \cos (bx - 4)\), we can rewrite it in the form \(a \cos (bx + \tilde{c})\), where \(\tilde{c}\) is a positive constant, by taking advantage of the periodic property of the cosine function and using \(\tilde{c} = 2k \pi + 4\), where \(k\) is an integer.