In right triangle LMN, right angled at M, if angle N = 60°, determine the exact value of tan L.

Introduction / Context:

This question tests your understanding of angle relationships in a right triangle and the standard values of trigonometric ratios. You are given one acute angle in a right triangle and asked to find the tangent of the other acute angle. Recognising that the two acute angles are complementary is the main idea here.

Given Data / Assumptions:

• Triangle LMN is right angled at M, so angle M = 90°.
• Angle N is 60°.
• We must find tan L.
• The sum of the three interior angles of a triangle is 180°.

Concept / Approach:

In any right triangle, the two acute angles add up to 90°. Therefore, if one acute angle is 60°, the other must be 30°. Once we identify angle L as 30°, we recall the standard value tan 30° = 1/√3. This gives the required value directly, without needing to work with side lengths explicitly.

Step-by-Step Solution:

Verification / Alternative check:

You can also check using known side ratios. In a 30°-60°-90° triangle, the sides are in the ratio 1 : √3 : 2, where the smallest side opposite 30° is 1, the side opposite 60° is √3, and the hypotenuse is 2. For angle L = 30°, tan L = opposite / adjacent = 1 / √3, which agrees with the special angle value and confirms our conclusion.

Why Other Options Are Wrong:

The values 1/2, 1/√2, 2, and √3 correspond to tan values of 26.565°, 45°, or 60°, but not 30°. In particular, tan 60° = √3 and tan 45° = 1, while tan 30° is distinctly 1/√3. Thus only option 1/√3 matches the correct tangent of angle L.

Common Pitfalls:

A frequent mistake is to assume tan L equals tan N or to confuse which angle is 30° and which is 60°. Some learners also mix up the standard values for tan 30° and tan 60°. Remembering the right triangle pattern and the special angle table helps avoid such confusion.

Final Answer:

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