Solve sin ` (cos ^(-1) x) = (1)/(9)`

Options

A. `(4sqrt5)/( 9)`
B. ` ( 2sqrt5)/( 9)`
C. ` ( 4sqrt3)/(5)`
D. ` (2sqrt3)/(5)`

Answer

To solve the equation \( \sin(\cos^{-1} x) = \frac{1}{9} \), we will follow these steps:

### Step 1: Set up the equation
Let \( \alpha = \cos^{-1} x \). Therefore, we can rewrite the equation as:
\[ \sin(\alpha) = \frac{1}{9} \]

### Step 2: Relate sine and cosine
From the definition of cosine, we know:
\[ \cos(\alpha) = x \]
We can visualize this using a right triangle where:
- The hypotenuse is 1 (the length of the hypotenuse in a unit circle),
- The base (adjacent side) is \( x \),
- The opposite side (perpendicular) can be determined using the Pythagorean theorem.

### Step 3: Use the Pythagorean theorem
Using the Pythagorean theorem:
\[ \text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2 \]
We have:
\[ \text{opposite}^2 + x^2 = 1^2 \]
Thus,
\[ \text{opposite}^2 = 1 - x^2 \]
Taking the square root gives us:
\[ \text{opposite} = \sqrt{1 - x^2} \]

### Step 4: Substitute into the sine equation
Now substituting back into the sine function, we have:
\[ \sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{1 - x^2}}{1} = \sqrt{1 - x^2} \]
Setting this equal to \( \frac{1}{9} \):
\[ \sqrt{1 - x^2} = \frac{1}{9} \]

### Step 5: Square both sides
To eliminate the square root, we square both sides:
\[ 1 - x^2 = \left(\frac{1}{9}\right)^2 \]
This simplifies to:
\[ 1 - x^2 = \frac{1}{81} \]

### Step 6: Solve for \( x^2 \)
Rearranging gives:
\[ x^2 = 1 - \frac{1}{81} \]
Finding a common denominator (81):
\[ x^2 = \frac{81}{81} - \frac{1}{81} = \frac{80}{81} \]

### Step 7: Take the square root
Taking the square root of both sides, we find:
\[ x = \sqrt{\frac{80}{81}} = \frac{\sqrt{80}}{9} \]
We can simplify \( \sqrt{80} \):
\[ \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} \]
Thus,
\[ x = \frac{4\sqrt{5}}{9} \]

### Final Answer
The solution to the equation \( \sin(\cos^{-1} x) = \frac{1}{9} \) is:
\[ x = \frac{4\sqrt{5}}{9} \]