Let `l= sin theta, m=cos theta and n=tan theta`.
If `theta=5` radian, then :

Options

A. `l gt m`
B. `l lt m`
C. `l =m`
D. none of these

Answer

To solve the problem step by step, we will calculate the values of \( l \), \( m \), and \( n \) given \( \theta = 5 \) radians.

### Step 1: Calculate \( l = \sin(\theta) \)
Given \( \theta = 5 \) radians, we find:
\[ l = \sin(5) \]
To simplify this, we can express \( 5 \) radians in terms of a known angle. We can write:
\[ 5 = 3\pi/2 + (5 - 3\pi/2) \]
This means:
\[ l = \sin(3\pi/2 + (5 - 3\pi/2)) \]
Using the sine addition formula, we know:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
where \( a = 3\pi/2 \) and \( b = 5 - 3\pi/2 \).

Now, since \( \sin(3\pi/2) = -1 \) and \( \cos(3\pi/2) = 0 \), we have:
\[ l = \sin(3\pi/2)\cos(5 - 3\pi/2) + \cos(3\pi/2)\sin(5 - 3\pi/2) \]
This simplifies to:
\[ l = -1 \cdot \cos(5 - 3\pi/2) + 0 \cdot \sin(5 - 3\pi/2) = -\cos(5 - 3\pi/2) \]

### Step 2: Calculate \( m = \cos(\theta) \)
Now we calculate:
\[ m = \cos(5) \]
Using the same approach:
\[ m = \cos(3\pi/2 + (5 - 3\pi/2)) \]
Again, applying the cosine addition formula:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
where \( a = 3\pi/2 \) and \( b = 5 - 3\pi/2 \).

Thus:
\[ m = \cos(3\pi/2)\cos(5 - 3\pi/2) - \sin(3\pi/2)\sin(5 - 3\pi/2) \]
This simplifies to:
\[ m = 0 \cdot \cos(5 - 3\pi/2) - (-1) \cdot \sin(5 - 3\pi/2) = \sin(5 - 3\pi/2) \]

### Step 3: Calculate \( n = \tan(\theta) \)
Now we find:
\[ n = \tan(5) = \frac{\sin(5)}{\cos(5)} = \frac{l}{m} \]

### Step 4: Compare \( l \) and \( m \)
From the previous calculations:
- \( l = -\cos(5 - 3\pi/2) \)
- \( m = \sin(5 - 3\pi/2) \)

Since \( l \) is negative and \( m \) is positive (as sine is positive in the fourth quadrant), we conclude:
\[ l < m \]

### Conclusion
Thus, the correct option is that \( l < m \).

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