The max. and min. values of `8 cos theta - 15 sin theta` are `"……………."` and `"………"`

Answer

To find the maximum and minimum values of the expression \(8 \cos \theta - 15 \sin \theta\), we can use the following steps:

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

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

### Step 3: Calculate \(a^2 + b^2\)
Now, we calculate \(a^2 + b^2\):
\[ a^2 = 8^2 = 64 \]
\[ b^2 = (-15)^2 = 225 \]
\[ a^2 + b^2 = 64 + 225 = 289 \]

### Step 4: Find the square root
Next, we find the square root of \(289\):
\[ \sqrt{289} = 17 \]

### Step 5: Determine the maximum and minimum values
Now we can use the values we calculated:
- Maximum value = \(\sqrt{a^2 + b^2} = 17\)
- Minimum value = \(-\sqrt{a^2 + b^2} = -17\)

### Final Answer
Thus, the maximum value of \(8 \cos \theta - 15 \sin \theta\) is \(17\) and the minimum value is \(-17\).

---