Find the difference between square of greatest and least value of `15 cos theta-8sin theta+5` .

Answer

To solve the problem, we need to find the difference between the square of the greatest and least value of the expression \(15 \cos \theta - 8 \sin \theta + 5\).

### Step 1: Identify the coefficients
The expression can be rewritten in the form \(a \cos \theta + b \sin \theta + c\), where:
- \(a = 15\)
- \(b = -8\)
- \(c = 5\)

### Step 2: Calculate the maximum and minimum values
The maximum and minimum values of the expression \(a \cos \theta + b \sin \theta\) can be calculated using the formula:
- Maximum value = \(\sqrt{a^2 + b^2}\)
- Minimum value = \(-\sqrt{a^2 + b^2}\)

Calculating \(a^2 + b^2\):
\[ a^2 + b^2 = 15^2 + (-8)^2 = 225 + 64 = 289 \]
Thus,
\[ \sqrt{a^2 + b^2} = \sqrt{289} = 17 \]

So, the maximum value of \(15 \cos \theta - 8 \sin \theta\) is \(17\), and the minimum value is \(-17\).

### Step 3: Include the constant term
Now, we add the constant \(c = 5\) to both the maximum and minimum values:
- Maximum value of the expression: \(17 + 5 = 22\)
- Minimum value of the expression: \(-17 + 5 = -12\)

### Step 4: Calculate the squares of the maximum and minimum values
Now, we need to find the squares of these values:
- Square of the maximum value: \(22^2 = 484\)
- Square of the minimum value: \((-12)^2 = 144\)

### Step 5: Find the difference between the squares
Now, we find the difference between the square of the greatest and least values:
\[ \text{Difference} = 484 - 144 = 340 \]

### Final Answer
The difference between the square of the greatest and least value of \(15 \cos \theta - 8 \sin \theta + 5\) is \(340\).

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