Maximum and minimum values of `15 cos theta-8 sin theta` are…......... and ….......

Answer

To find the maximum and minimum values of the expression \( 15 \cos \theta - 8 \sin \theta \), 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:
- \( a = 15 \)
- \( b = -8 \)

### Step 2: Calculate \( \sqrt{a^2 + b^2} \)
To find the maximum and minimum values, we first calculate \( \sqrt{a^2 + b^2} \):
\[ a^2 = 15^2 = 225 \]
\[ b^2 = (-8)^2 = 64 \]
\[ a^2 + b^2 = 225 + 64 = 289 \]
\[ \sqrt{a^2 + b^2} = \sqrt{289} = 17 \]

### Step 3: Determine the maximum and minimum values
The maximum value of the expression \( 15 \cos \theta - 8 \sin \theta \) is given by:
\[ \text{Maximum value} = \sqrt{a^2 + b^2} = 17 \]

The minimum value is given by:
\[ \text{Minimum value} = -\sqrt{a^2 + b^2} = -17 \]

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

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