Question

How do you find the maximum values of the function $f\left( x \right) = 5\sin x + 7\cos x$?

Answer 1

In the given question, we are provided with a trigonometric expression involving the trigonometric functions sine and cosine and we are required to find the maximum value of the expression. To solve the problem, we must know the technique of calculating the range of the trigonometric expression $\left( {a\sin x + b\cos x} \right)$. We first divide such expressions by $\sqrt {{a^2} + {b^2}}$ to calculate the range and then transform the sum of two trigonometric functions into a single term.

Complete step by step solution:
The given function is $f\left( x \right) = 5\sin x + 7\cos x$.
We have to find the range of the trigonometric expression $\left( {a\sin x + b\cos x} \right)$.
We divide such a trigonometric expression by $\sqrt {{a^2} + {b^2}}$ to transform the sum of two trigonometric ratios into one term.
So, we have, $f\left( x \right) = 5\sin x + 7\cos x$
We divide and multiply the expression by $\sqrt {{5^2} + {7^2}} = \sqrt {25 + 49} = \sqrt {74}$. Hence, we get,
$\Rightarrow f\left( x \right) = \sqrt {74} \left( {\dfrac{5}{{\sqrt {74} }}\sin x + \dfrac{7}{{\sqrt {74} }}\cos x} \right)$
Now, we can assume $\dfrac{5}{{\sqrt {74} }}$ as a cosine of some angle, say y.
Then, $\dfrac{7}{{\sqrt {74} }}$ will correspond to the sine of same angle y as we know that ${\sin ^2}x + {\cos ^2}x = 1$ and ${\left( {\dfrac{5}{{\sqrt {74} }}} \right)^2} + {\left( {\dfrac{7}{{\sqrt {74} }}} \right)^2} = 1$.
Hence, we have, $\sin y = \left( {\dfrac{7}{{\sqrt {74} }}} \right)$ and $\cos y = \left( {\dfrac{5}{{\sqrt {74} }}} \right)$.
Substituting the values, we get,
$\Rightarrow f\left( x \right) = \sqrt {74} \left( {\sin x\cos y + \sin y\cos x} \right)$
Now, we know the compound angle formula for sine as $\left( {\cos \theta \sin \phi + \sin \theta \cos \phi } \right) = \sin \left( {\phi + \theta } \right)$. Hence, we get,
$\Rightarrow f\left( x \right) = \sqrt {74} \sin \left( {x + y} \right)$
Now, we know that sine of any angle has a range of $\left[ { - 1,1} \right]$. So, the maximum value of sine function can be one.
Hence, the maximum range of $\sqrt {74} \sin \left( {x + y} \right)$ is $\sqrt {74}$.
Therefore, the maximum value of $f\left( x \right) = 5\sin x + 7\cos x$ is $\sqrt {74}$.

Note:
Such questions require grip over the concepts of trigonometry and inequalities. One must know the methodology to calculate the range of trigonometric expressions of the form $\left( {a\sin x + b\cos x} \right)$ in order to solve the given problem. Dividing or multiplying any inequality by a positive number does not change the signs of the inequality. But when we multiply or divide any inequality by a negative number, the signs of the inequality are reversed. Whereas, in the case of the equation, both sides remain equal if multiplied or divided by a positive or negative number.