Questions: Determine the minimum and maximum value of the following trigonometric function. f(x)=4 sin x+3

Determine the minimum and maximum value of the following trigonometric function. f(x)=4 sin x+3
Transcript text: Determine the minimum and maximum value of the following trigonometric function. \[ f(x)=4 \sin x+3 \]
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Solution

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Solution Steps

Step 1: Identify the Range of the Sine Function

The sine function, \(\sin x\), has a range of \([-1, 1]\). This means that the values of \(\sin x\) will always lie between \(-1\) and \(1\).

Step 2: Apply the Range to the Given Function

The given function is:

\[ f(x) = 4 \sin x + 3 \]

To find the range of \(f(x)\), we need to apply the range of \(\sin x\) to this function.

Step 3: Calculate the Minimum Value

The minimum value of \(\sin x\) is \(-1\). Substituting this into the function:

\[ f(x)_{\text{min}} = 4(-1) + 3 = -4 + 3 = -1 \]

Step 4: Calculate the Maximum Value

The maximum value of \(\sin x\) is \(1\). Substituting this into the function:

\[ f(x)_{\text{max}} = 4(1) + 3 = 4 + 3 = 7 \]

Final Answer

The minimum value of the function \(f(x) = 4 \sin x + 3\) is \(\boxed{-1}\), and the maximum value is \(\boxed{7}\).

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