(A): The method of dimensions cannot be used to obtain the dependence of work done by a force when the force is inclined to the direction of displacement.
(R): All trigonometric functions are dimensionless.

Options

A. Both (A) and (R) are true and (R) is the correct explanation of (A)
B. Both (A) and (R) are true and (R) is not the correct explanation of (A)
C. (A) is true but (R) is false
D. Both (A) and (R) are false

Answer

To solve the question, we need to analyze the statements (A) and (R) and determine their validity.

### Step-by-Step Solution:

1. **Understanding Statement (A)**:
- Statement (A) claims that the method of dimensions cannot be used to obtain the dependence of work done by a force when the force is inclined to the direction of displacement.
- Work done (W) is given by the equation \( W = F \cdot s \cdot \cos(\theta) \), where \( F \) is the force, \( s \) is the displacement, and \( \theta \) is the angle between the force and the displacement.

2. **Analyzing the Equation**:
- The equation for work includes a trigonometric function \( \cos(\theta) \).
- Trigonometric functions are dimensionless, meaning they do not have any units associated with them.

3. **Using Dimensional Analysis**:
- Dimensional analysis is a technique used to derive relationships between physical quantities based on their dimensions.
- However, it cannot determine constants or the effects of dimensionless quantities like trigonometric functions.

4. **Conclusion for Statement (A)**:
- Since the work done depends on the angle \( \theta \) through the cosine function, and since dimensional analysis cannot account for the effects of this angle, we conclude that statement (A) is true.

5. **Understanding Statement (R)**:
- Statement (R) states that all trigonometric functions are dimensionless.
- This is a fundamental property of trigonometric functions, as they are ratios of lengths (sides of triangles), which cancel out their dimensions.

6. **Conclusion for Statement (R)**:
- Since it is indeed true that all trigonometric functions are dimensionless, statement (R) is also true.

7. **Final Evaluation**:
- Both statements (A) and (R) are true. Statement (R) provides a valid explanation for why statement (A) is true.

### Final Answer:
- Both statements (A) and (R) are true, and (R) is the correct explanation for (A).

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