If `x in (0, 1)`, then greatest root of the equation `sin2pix=sqrt2cospix` is

Options

A. `1/4`
B. `1/2`
C. `3/4`
D. 1/2

Answer

To solve the equation \( \sin(2\pi x) = \sqrt{2} \cos(\pi x) \) for \( x \) in the interval \( (0, 1) \), we can follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that \( \sin(2a) = 2 \sin(a) \cos(a) \). Therefore, we can rewrite \( \sin(2\pi x) \) as:
\[ \sin(2\pi x) = 2 \sin(\pi x) \cos(\pi x) \]
Thus, the equation becomes:
\[ 2 \sin(\pi x) \cos(\pi x) = \sqrt{2} \cos(\pi x) \]

### Step 2: Rearrange the equation
We can rearrange the equation to isolate the terms:
\[ 2 \sin(\pi x) \cos(\pi x) - \sqrt{2} \cos(\pi x) = 0 \]

### Step 3: Factor out common terms
We can factor out \( \cos(\pi x) \):
\[ \cos(\pi x) (2 \sin(\pi x) - \sqrt{2}) = 0 \]

### Step 4: Set each factor to zero
This gives us two cases to consider:
1. \( \cos(\pi x) = 0 \)
2. \( 2 \sin(\pi x) - \sqrt{2} = 0 \)

### Step 5: Solve for \( \cos(\pi x) = 0 \)
The cosine function is zero at odd multiples of \( \frac{\pi}{2} \):
\[ \pi x = \frac{\pi}{2} \implies x = \frac{1}{2} \]

### Step 6: Solve for \( 2 \sin(\pi x) - \sqrt{2} = 0 \)
Rearranging gives:
\[ 2 \sin(\pi x) = \sqrt{2} \implies \sin(\pi x) = \frac{\sqrt{2}}{2} \]
The sine function equals \( \frac{\sqrt{2}}{2} \) at:
\[ \pi x = \frac{\pi}{4} \implies x = \frac{1}{4} \]
And also at:
\[ \pi x = \frac{3\pi}{4} \implies x = \frac{3}{4} \]

### Step 7: List all roots found
The roots we have found are:
- \( x = \frac{1}{2} \)
- \( x = \frac{1}{4} \)
- \( x = \frac{3}{4} \)

### Step 8: Determine the greatest root
Among the roots \( \frac{1}{2}, \frac{1}{4}, \frac{3}{4} \), the greatest root is:
\[ \frac{3}{4} \]

### Final Answer
The greatest root of the equation \( \sin(2\pi x) = \sqrt{2} \cos(\pi x) \) for \( x \in (0, 1) \) is:
\[ \boxed{\frac{3}{4}} \]