Find the maximum and minimum values of `8costheta+15sintheta`:

Options

A. 17 and -17
B. 16 and -9
C. 25 and 0
D. 36 and 25

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 coefficients
In the expression \( 8 \cos \theta + 15 \sin \theta \), we identify:
- \( a = 15 \)
- \( b = 8 \)

### Step 2: Use the formula for maximum and minimum values
The maximum and minimum values of the expression \( a \sin \theta + b \cos \theta \) 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 \( a^2 + b^2 \):
\[ a^2 = 15^2 = 225 \]
\[ b^2 = 8^2 = 64 \]
\[ a^2 + b^2 = 225 + 64 = 289 \]

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

### Step 5: Determine maximum and minimum values
Now we can determine the maximum and minimum values:
- Maximum value = \( 17 \)
- Minimum value = \( -17 \)

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

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