In right triangle LMN, right angled at M, you are told that ∠N = 60°. Use the property that the acute angles are complementary to find the exact value of tan L and select the correct option.

Introduction / Context:
This trigonometry question tests your understanding of complementary angles in a right triangle. When one angle in a right triangle is known, you can always find the other acute angle, since their sum must be 90 degrees. Using basic trigonometric values for 30 degrees and 60 degrees, you can then evaluate the required ratio, in this case tan L, without needing any complex computation.

Given Data / Assumptions:

• Triangle LMN is right angled at M, so ∠M = 90 degrees.
• ∠N = 60 degrees.
• The sum of angles in a triangle is 180 degrees.
• tan θ is defined as opposite side / adjacent side for an acute angle θ in a right triangle.
• We work with standard exact values for 30 degrees and 60 degrees.

Concept / Approach:
In a right triangle, the two acute angles are complementary. That is, if one acute angle is 60 degrees, the other must be 30 degrees. Here, since ∠N = 60 degrees and ∠M = 90 degrees, angle L must be 30 degrees. Once we establish that L is 30 degrees, we can simply recall that tan 30 degrees has the exact value 1/√3, which is one of the standard special-angle trigonometric ratios.

Step-by-Step Solution:
Use the angle sum property: ∠L + ∠M + ∠N = 180 degrees. Substitute the known values: ∠L + 90 + 60 = 180, so ∠L + 150 = 180. Solve for ∠L: ∠L = 180 - 150 = 30 degrees. Now we need tan L, which is tan 30 degrees. Recall the standard value: tan 30 degrees = 1/√3.
Verification / Alternative check:
You can think of a standard 30–60–90 triangle, where the sides are in the ratio 1 : √3 : 2. For the 30 degree angle, opposite side is 1 and adjacent side is √3, so tan 30 degrees = 1/√3. For the 60 degree angle, the roles reverse and tan 60 degrees = √3. This confirms that the correct value for tan L (which is 30 degrees) is 1/√3.

Why Other Options Are Wrong:
Option b (1/2) is the value of sin 30 degrees, not tan 30 degrees. Option c (1/√2) is often associated with sin 45 degrees or cos 45 degrees, and does not apply here. Option d (2) is not a standard tangent value for 30 or 60 degrees. Option e (√3) is tan 60 degrees, which would be the tangent of angle N, not angle L.

Common Pitfalls:
Learners sometimes mix up which angle is 30 degrees and which is 60 degrees in the right triangle, especially when not drawing a diagram. Another common error is to recall tan 30 degrees as √3 or tan 60 degrees as 1/√3, effectively swapping them. Carefully identifying that L is the smaller acute angle and remembering the standard triangle with side ratio 1 : √3 : 2 helps avoid these mistakes.

Final Answer:
The correct value of tan L in the given right triangle is 1/√3.