Find the range of `5 sin x – 12 cos x + 7`

Answer

To find the range of the expression \(5 \sin x - 12 \cos x + 7\), we can follow these steps:

### Step 1: Identify the coefficients
We have the expression in the form \(a \sin x + b \cos x\), where:
- \(a = 5\)
- \(b = -12\)

### Step 2: Calculate the amplitude
The range of \(a \sin x + b \cos x\) can be determined using the formula:
\[ \text{Range} = -\sqrt{a^2 + b^2} \text{ to } \sqrt{a^2 + b^2} \]
First, we calculate \(a^2 + b^2\):
\[ a^2 = 5^2 = 25 \]
\[ b^2 = (-12)^2 = 144 \]
\[ a^2 + b^2 = 25 + 144 = 169 \]

### Step 3: Find the square root
Now, we find the square root of \(169\):
\[ \sqrt{169} = 13 \]

### Step 4: Determine the range of \(5 \sin x - 12 \cos x\)
Using the results from the previous steps, we find the range of \(5 \sin x - 12 \cos x\):
\[ \text{Range} = -13 \text{ to } 13 \]

### Step 5: Shift the range by adding 7
Now, we need to find the range of the entire expression \(5 \sin x - 12 \cos x + 7\). We shift the range by adding \(7\):
\[ \text{New Range} = -13 + 7 \text{ to } 13 + 7 \]
Calculating the new limits:
\[ -13 + 7 = -6 \]
\[ 13 + 7 = 20 \]

### Final Result
Thus, the range of the expression \(5 \sin x - 12 \cos x + 7\) is:
\[ \text{Range} = [-6, 20] \]

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