Finding the Maximum Value of \(a\cos x + b \sin x + c\)
We want to determine the maximum possible value for the expression \(a\cos x + b \sin x + c\). This involves understanding the range of the trigonometric part, \(a\cos x + b \sin x\).
Understanding the Trigonometric Component
Consider the term \(a\cos x + b \sin x\). We can rewrite this expression in the form \(R\cos(x - \alpha)\), where \(R > 0\). Using the angle subtraction identity for cosine, we have:
\(R\cos(x - \alpha) = R(\cos x \cos \alpha + \sin x \sin \alpha)\)
\(= (R\cos \alpha)\cos x + (R\sin \alpha)\sin x\)
By comparing this with \(a\cos x + b \sin x\), we can equate the coefficients:
- \(a = R\cos \alpha\)
- \(b = R\sin \alpha\)
To find \(R\), we can square and add these two equations:
\(a^2 + b^2 = (R\cos \alpha)^2 + (R\sin \alpha)^2\)
\(a^2 + b^2 = R^2\cos^2 \alpha + R^2\sin^2 \alpha\)
\(a^2 + b^2 = R^2(\cos^2 \alpha + \sin^2 \alpha)\)
Since \(\cos^2 \alpha + \sin^2 \alpha = 1\), we get:
\(a^2 + b^2 = R^2\)
As \(R > 0\), we find \(R = \sqrt{a^2 + b^2}\).
Therefore, the expression \(a\cos x + b \sin x\) can be written as \(\sqrt{a^2 + b^2}\cos(x - \alpha)\) for some angle \(\alpha\).
Determining the Maximum Value
The cosine function, \(\cos(\theta)\), has a maximum value of 1 and a minimum value of -1, regardless of the angle \(\theta\). In our case, \(\theta = x - \alpha\).
So, the maximum value of \(\cos(x - \alpha)\) is 1.
Consequently, the maximum value of \(\sqrt{a^2 + b^2}\cos(x - \alpha)\) is \(\sqrt{a^2 + b^2} \times 1 = \sqrt{a^2 + b^2}\).
Now, let's consider the full expression: \(a\cos x + b \sin x + c\). We found the maximum value of \(a\cos x + b \sin x\) is \(\sqrt{a^2 + b^2}\).
To find the maximum value of the entire expression, we simply add the constant \(c\) to the maximum value of the trigonometric part:
Maximum value = (Maximum value of \(a\cos x + b \sin x\)) + \(c\)
Maximum value = \(\sqrt{a^2 + b^2} + c\).
Conclusion
The maximum value of the expression \(a\cos x + b \sin x + c\) is \(\sqrt{a^2 + b^2} + c\). This corresponds to option 2.
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