If `cos x = (3)/(5)` , then find the value of `sin x - sin^(3) x `.

Options

A. `0.476`
B. `0.288`
C. `0.358`
D. `0.389`

Answer

To solve the problem where \( \cos x = \frac{3}{5} \) and we need to find \( \sin x - \sin^3 x \), we can follow these steps:

### Step 1: Understand the relationship between sine and cosine
We know that:
\[ \sin^2 x + \cos^2 x = 1 \]
Given \( \cos x = \frac{3}{5} \), we can find \( \sin^2 x \).

### Step 2: Calculate \( \sin^2 x \)
Substituting the value of \( \cos x \) into the Pythagorean identity:
\[ \sin^2 x + \left(\frac{3}{5}\right)^2 = 1 \]
\[ \sin^2 x + \frac{9}{25} = 1 \]
\[ \sin^2 x = 1 - \frac{9}{25} \]
\[ \sin^2 x = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]

### Step 3: Find \( \sin x \)
Taking the square root of both sides, we get:
\[ \sin x = \sqrt{\frac{16}{25}} = \frac{4}{5} \]
(Note: We take the positive root since sine is positive in the first quadrant where cosine is positive.)

### Step 4: Calculate \( \sin^3 x \)
Now we calculate \( \sin^3 x \):
\[ \sin^3 x = \left(\frac{4}{5}\right)^3 = \frac{64}{125} \]

### Step 5: Substitute into the expression \( \sin x - \sin^3 x \)
Now we substitute \( \sin x \) and \( \sin^3 x \) into the expression:
\[ \sin x - \sin^3 x = \frac{4}{5} - \frac{64}{125} \]

### Step 6: Find a common denominator
The common denominator between 5 and 125 is 125. We convert \( \frac{4}{5} \):
\[ \frac{4}{5} = \frac{4 \times 25}{5 \times 25} = \frac{100}{125} \]

### Step 7: Perform the subtraction
Now we can perform the subtraction:
\[ \sin x - \sin^3 x = \frac{100}{125} - \frac{64}{125} = \frac{100 - 64}{125} = \frac{36}{125} \]

### Final Answer
Thus, the value of \( \sin x - \sin^3 x \) is:
\[ \frac{36}{125} \]