Find the range of `12sintheta-9sin^2theta`

Explanation

To find the range of the expression \( 12\sin\theta - 9\sin^2\theta \), we can follow these steps:

### Step 1: Rewrite the expression
Let \( y = 12\sin\theta - 9\sin^2\theta \). We can rearrange this into a standard quadratic form in terms of \( \sin\theta \):
\[ y = -9\sin^2\theta + 12\sin\theta \]

### Step 2: Identify the quadratic coefficients
This is a quadratic equation in the form \( ay^2 + by + c \), where:
- \( a = -9 \)
- \( b = 12 \)
- \( c = 0 \)

### Step 3: Find the vertex of the quadratic
The maximum or minimum value of a quadratic equation \( ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \). Here, we need to find \( \sin\theta \):
\[ \sin\theta = -\frac{12}{2 \times -9} = \frac{12}{18} = \frac{2}{3} \]

### Step 4: Calculate the maximum value
Substituting \( \sin\theta = \frac{2}{3} \) back into the expression to find the maximum value:
\[ y = 12\left(\frac{2}{3}\right) - 9\left(\frac{2}{3}\right)^2 \]
Calculating each term:
\[ y = 12 \times \frac{2}{3} - 9 \times \frac{4}{9} \]
\[ y = 8 - 4 = 4 \]

### Step 5: Find the minimum value
To find the minimum value, we need to consider the extreme values of \( \sin\theta \), which are \( -1 \) and \( 1 \).

1. For \( \sin\theta = -1 \):
\[ y = 12(-1) - 9(-1)^2 = -12 - 9 = -21 \]

2. For \( \sin\theta = 1 \):
\[ y = 12(1) - 9(1)^2 = 12 - 9 = 3 \]

### Step 6: Determine the range
From the calculations:
- Maximum value = \( 4 \)
- Minimum value = \( -21 \)

Thus, the range of the expression \( 12\sin\theta - 9\sin^2\theta \) is:
\[ [-21, 4] \]