`triangleLMN` is right angled at M. If `M - N=60^(@)," then tan L="`

Options

A. `√
3
`
B. `1/sqrt3`
C. `1/sqrt2`
D. 2

Answer

To solve the problem, we will follow these steps:

1. **Understanding the Triangle**: We have a right-angled triangle LMN with the right angle at M. This means that angle M = 90°.

2. **Using the Given Information**: We know that M - N = 60°. Since M = 90°, we can substitute this into the equation:
\[ 90° - N = 60° \]

3. **Solving for Angle N**: Rearranging the equation to find angle N:
\[ N = 90° - 60° = 30° \]

4. **Finding Angle L**: The sum of angles in a triangle is 180°. Therefore, we can find angle L using the equation:
\[ L + M + N = 180° \]
Substituting the known values:
\[ L + 90° + 30° = 180° \]
Simplifying this gives:
\[ L + 120° = 180° \]
Therefore:
\[ L = 180° - 120° = 60° \]

5. **Calculating tan L**: Now, we need to find tan L. Since we have found that angle L = 60°, we can use the trigonometric identity:
\[ \tan(60°) = \sqrt{3} \]

Thus, the final answer is:
\[ \tan L = \sqrt{3} \]