Find `max^(m)&min^(m)` value of `8sintheta+15costheta`.

Answer

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

### Step 1: Identify the coefficients
The coefficients of \(\sin \theta\) and \(\cos \theta\) are:
- \(a = 8\)
- \(b = 15\)

### Step 2: Use the formula for maximum and minimum values
The maximum value of the expression \(a \sin \theta + b \cos \theta\) is given by:
\[ \text{Max} = \sqrt{a^2 + b^2} \]
The minimum value is given by:
\[ \text{Min} = -\sqrt{a^2 + b^2} \]

### 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: Calculate the square root
Next, we find the square root:
\[ \sqrt{a^2 + b^2} = \sqrt{289} = 17 \]

### Step 5: Determine the maximum and minimum values
Now we can state the maximum and minimum values:
- Maximum value:
\[ \text{Max} = 17 \]
- Minimum value:
\[ \text{Min} = -17 \]

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

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