If `f (theta) = |sin theta| + |cos theta|, theta in R`, then

Options

A. `f(theta) in [0, 2]`
B. `f(theta)in [0, sqrt(2)]`
C. `f(theta) in [ 0, 1]`
D. `f(theta) in [1, sqrt(2)]`

Answer

To find the range of the function \( f(\theta) = |\sin \theta| + |\cos \theta| \), we can follow these steps:

### Step 1: Rewrite the Function
We start with the function:
\[ f(\theta) = |\sin \theta| + |\cos \theta| \]

### Step 2: Use Trigonometric Identities
We can multiply and divide the function by \( \sqrt{2} \):
\[ f(\theta) = \frac{1}{\sqrt{2}} |\sin \theta| + \frac{1}{\sqrt{2}} |\cos \theta| \cdot \sqrt{2} \]

### Step 3: Recognize Cosine and Sine Values
Notice that \( \frac{1}{\sqrt{2}} = \cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) \). Thus, we can rewrite the function as:
\[ f(\theta) = \sqrt{2} \left( \cos\left(\frac{\pi}{4}\right) |\sin \theta| + \sin\left(\frac{\pi}{4}\right) |\cos \theta| \right) \]

### Step 4: Apply the Sine Addition Formula
Using the sine addition formula \( \sin(a + b) = \sin a \cos b + \cos a \sin b \), we can express the function as:
\[ f(\theta) = \sqrt{2} \left| \sin\left(\theta + \frac{\pi}{4}\right) \right| \]

### Step 5: Determine the Range of the Sine Function
The sine function \( \sin x \) has a range of \([-1, 1]\). Therefore, the absolute value \( |\sin x| \) has a range of \([0, 1]\).

### Step 6: Scale the Range
Since we have \( f(\theta) = \sqrt{2} |\sin\left(\theta + \frac{\pi}{4}\right)| \), the range of \( f(\theta) \) will be:
\[ [0, \sqrt{2}] \]

### Conclusion
Thus, the range of the function \( f(\theta) = |\sin \theta| + |\cos \theta| \) is:
\[ [0, \sqrt{2}] \]

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