Find the maximum possible value of $3\cos(x) + 4\sin(x)$.
Understand the Problem
The question asks us to find the maximum possible value of the expression $3\cos(x) + 4\sin(x)$. This can be solved by recognizing that the expression can be rewritten in the form $R\cos(x - \alpha)$ or $R\sin(x + \beta)$, where $R$ is the amplitude and represents the maximum value of the expression. We can find $R$ using the formula $R = \sqrt{a^2 + b^2}$, where $a$ and $b$ are the coefficients of $\cos(x)$ and $\sin(x)$, respectively.
Answer
$5$
Answer for screen readers
$5$
Steps to Solve
- Identify the coefficients
The given expression is $3\cos(x) + 4\sin(x)$. We identify the coefficient of $\cos(x)$ as $a = 3$ and the coefficient of $\sin(x)$ as $b = 4$.
- Calculate R using the formula
The maximum possible value of the expression is given by $R = \sqrt{a^2 + b^2}$. Substituting the values of $a$ and $b$, we get: $$R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$
- State the maximum value
The maximum possible value of the expression $3\cos(x) + 4\sin(x)$ is $R = 5$.
$5$
More Information
The expression $3\cos(x) + 4\sin(x)$ can be rewritten in the form $5\cos(x - \alpha)$ or $5\sin(x + \beta)$ where $\alpha = \arctan(\frac{4}{3})$ and $\beta = \arctan(\frac{3}{4})$. The maximum value of the cosine or sine function is 1, which occurs when $x = \alpha$ in the cosine form or $x = \frac{\pi}{2} - \beta$ in the sine form, thus confirming the maximum value of the expression to be 5.
Tips
A common mistake is forgetting to take the square root after summing the squares of the coefficients. Another mistake could be incorrectly identifying the coefficients of $\cos(x)$ and $\sin(x)$. Ensure correct substitution into the formula $R = \sqrt{a^2 + b^2}$.
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