The value of `sin1.cos2.tan3.cot4.sec5.cosec6` is

Options

A. Positive
B. Negative
C. Zero
D. May be positive and Negative

Answer

To solve the problem of finding the value of \( \sin(1) \cdot \cos(2) \cdot \tan(3) \cdot \cot(4) \cdot \sec(5) \cdot \csc(6) \), we will analyze each trigonometric function step by step, considering that the angles are in radians.

### Step-by-Step Solution:

1. **Identify the angles in radians**:
- The angles are \(1\), \(2\), \(3\), \(4\), \(5\), and \(6\) radians.

2. **Determine the quadrant for each angle**:
- **Angle 1 radian**:
- \(1\) radian is approximately \(57.29^\circ\), which is in the first quadrant.
- In the first quadrant, all trigonometric functions are positive.
- **Angle 2 radians**:
- \(2\) radians is approximately \(114.58^\circ\), which is in the second quadrant.
- In the second quadrant, sine is positive and cosine is negative.
- **Angle 3 radians**:
- \(3\) radians is approximately \(171.87^\circ\), which is also in the second quadrant.
- Therefore, tangent (which is sine over cosine) is negative.
- **Angle 4 radians**:
- \(4\) radians is approximately \(229.16^\circ\), which is in the third quadrant.
- In the third quadrant, cotangent (which is cosine over sine) is positive.
- **Angle 5 radians**:
- \(5\) radians is approximately \(286.45^\circ\), which is in the fourth quadrant.
- In the fourth quadrant, secant (which is the reciprocal of cosine) is positive.
- **Angle 6 radians**:
- \(6\) radians is approximately \(343.74^\circ\), which is also in the fourth quadrant.
- Therefore, cosecant (which is the reciprocal of sine) is negative.

3. **Evaluate the signs of each function**:
- \( \sin(1) \): Positive (1st quadrant)
- \( \cos(2) \): Negative (2nd quadrant)
- \( \tan(3) \): Negative (2nd quadrant)
- \( \cot(4) \): Positive (3rd quadrant)
- \( \sec(5) \): Positive (4th quadrant)
- \( \csc(6) \): Negative (4th quadrant)

4. **Combine the signs**:
- The product of the signs is:
- Positive (from \( \sin(1) \))
- Negative (from \( \cos(2) \))
- Negative (from \( \tan(3) \))
- Positive (from \( \cot(4) \))
- Positive (from \( \sec(5) \))
- Negative (from \( \csc(6) \))

- Thus, the overall sign can be calculated as:
- Positive \(\times\) Negative \(\times\) Negative \(\times\) Positive \(\times\) Positive \(\times\) Negative
- This simplifies to:
- Positive \(\times\) Negative \(\times\) Negative = Positive
- Positive \(\times\) Positive = Positive
- Positive \(\times\) Negative = Negative

5. **Final Result**:
- Therefore, the value of \( \sin(1) \cdot \cos(2) \cdot \tan(3) \cdot \cot(4) \cdot \sec(5) \cdot \csc(6) \) is **negative**.