If `x ge 0` and arcsin x = arccos (2x), then x =

Options

A. `0.866`
B. `0.707`
C. `0.500`
D. `0.447`

Explanation

To solve the equation \( \arcsin x = \arccos (2x) \) under the condition \( x \geq 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \arcsin x = \arccos (2x) \]
To eliminate the inverse trigonometric functions, we apply the sine function to both sides:
\[ \sin(\arcsin x) = \sin(\arccos (2x)) \]
This simplifies to:
\[ x = \sin(\arccos (2x)) \]

### Step 2: Use the identity for sine of arccosine
Using the identity \( \sin(\arccos y) = \sqrt{1 - y^2} \), we can rewrite the right side:
\[ x = \sqrt{1 - (2x)^2} \]
This simplifies to:
\[ x = \sqrt{1 - 4x^2} \]

### Step 3: Square both sides
Next, we square both sides to eliminate the square root:
\[ x^2 = 1 - 4x^2 \]

### Step 4: Rearrange the equation
Now, we rearrange the equation to bring all terms to one side:
\[ x^2 + 4x^2 = 1 \]
This simplifies to:
\[ 5x^2 = 1 \]

### Step 5: Solve for \( x^2 \)
Now, we divide both sides by 5:
\[ x^2 = \frac{1}{5} \]

### Step 6: Take the square root
Taking the square root of both sides gives us:
\[ x = \pm \frac{1}{\sqrt{5}} \]
Since \( x \geq 0 \), we discard the negative solution:
\[ x = \frac{1}{\sqrt{5}} \]

### Step 7: Approximate the value
Calculating the approximate value:
\[ x \approx 0.447 \]

### Conclusion
Thus, the solution to the equation \( \arcsin x = \arccos (2x) \) under the condition \( x \geq 0 \) is:
\[ \boxed{0.447} \]