Consider `S = X cos (theta)` for `X = (2.0 pm 0.2) cm, theta = 53 pm 2^(@)`. Find S.

Explanation

To solve the problem, we need to find the value of \( S \) given by the equation \( S = X \cos(\theta) \), where \( X = 2.0 \pm 0.2 \) cm and \( \theta = 53^\circ \pm 2^\circ \).

### Step-by-Step Solution:

1. **Calculate \( S \) without considering the error**:
\[ S = X \cos(\theta) \]
Substitute the values of \( X \) and \( \theta \):
\[ S = 2.0 \cos(53^\circ) \]
We know that \( \cos(53^\circ) \) is approximately \( \frac{3}{5} \) or \( 0.6 \):
\[ S = 2.0 \times 0.6 = 1.2 \text{ cm} \]

2. **Calculate the relative error in \( S \)**:
The formula for the propagation of uncertainty for a function of multiple variables is:
\[ \frac{\Delta S}{S} = \frac{\Delta X}{X} + \frac{\Delta (\cos \theta)}{\cos \theta} \]

3. **Calculate \( \Delta X \)**:
Given \( \Delta X = 0.2 \) cm and \( X = 2.0 \) cm:
\[ \frac{\Delta X}{X} = \frac{0.2}{2.0} = 0.1 \]

4. **Calculate \( \Delta (\cos \theta) \)**:
We need to find \( \Delta (\cos \theta) \) using the derivative:
\[ \Delta (\cos \theta) = -\sin(\theta) \Delta \theta \]
Here, \( \Delta \theta = 2^\circ \) (convert to radians):
\[ \Delta \theta = 2 \times \frac{\pi}{180} \approx 0.0349 \text{ radians} \]
Now, calculate \( \sin(53^\circ) \):
\[ \sin(53^\circ) \approx 0.7986 \]
Thus,
\[ \Delta (\cos \theta) = -0.7986 \times 0.0349 \approx -0.0279 \]
Now, calculate \( \frac{\Delta (\cos \theta)}{\cos \theta} \):
\[ \cos(53^\circ) \approx 0.6 \]
Therefore,
\[ \frac{\Delta (\cos \theta)}{\cos \theta} \approx \frac{-0.0279}{0.6} \approx -0.0465 \]

5. **Combine the relative errors**:
\[ \frac{\Delta S}{S} = 0.1 - 0.0465 \approx 0.0535 \]

6. **Calculate \( \Delta S \)**:
Now, calculate \( \Delta S \):
\[ \Delta S = S \times \frac{\Delta S}{S} = 1.2 \times 0.0535 \approx 0.0642 \text{ cm} \]

7. **Final result**:
The final result for \( S \) with its uncertainty is:
\[ S = 1.2 \pm 0.0642 \text{ cm} \]

### Final Answer:
\[ S = 1.2 \pm 0.064 \text{ cm} \]