The value of `cos 10^@ - sin 10^@` is

Options

A. Positive
B. Negative
C. 0
D. 1

Answer

To find the value of \( \cos 10^\circ - \sin 10^\circ \), we can analyze the functions involved and their properties.

### Step-by-Step Solution:

1. **Understanding the Functions**:
We know that \( \cos \theta \) and \( \sin \theta \) are trigonometric functions that vary with the angle \( \theta \). The cosine function starts at 1 when \( \theta = 0^\circ \) and decreases to 0 at \( \theta = 90^\circ \). The sine function starts at 0 when \( \theta = 0^\circ \) and increases to 1 at \( \theta = 90^\circ \).

**Hint**: Remember the basic properties of sine and cosine functions within the first quadrant (0° to 90°).

2. **Graphical Representation**:
If we plot the graphs of \( \cos \theta \) and \( \sin \theta \), we can observe that:
- At \( \theta = 0^\circ \), \( \cos 0^\circ = 1 \) and \( \sin 0^\circ = 0 \).
- At \( \theta = 45^\circ \), \( \cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2} \).
- For angles between \( 0^\circ \) and \( 45^\circ \), \( \cos \theta \) is greater than \( \sin \theta \).

**Hint**: Sketch or visualize the graphs of \( \cos \theta \) and \( \sin \theta \) to see their behavior in the first quadrant.

3. **Evaluating at \( 10^\circ \)**:
Since \( 10^\circ \) is less than \( 45^\circ \), we can conclude that:
\[ \cos 10^\circ > \sin 10^\circ \]

**Hint**: Use the fact that both functions are continuous and monotonic in the interval \( [0^\circ, 45^\circ] \).

4. **Conclusion**:
Therefore, we can state:
\[ \cos 10^\circ - \sin 10^\circ > 0 \]
This means that the value of \( \cos 10^\circ - \sin 10^\circ \) is positive.

**Hint**: Compare the values of \( \cos 10^\circ \) and \( \sin 10^\circ \) using a calculator or trigonometric tables if needed.

### Final Answer:
The value of \( \cos 10^\circ - \sin 10^\circ \) is positive.