In triangle PQR, angle Q is 90°. If m∠R = 30°, then using angle sum in a triangle, what is the exact value of cot P?

Introduction / Context: This question uses basic properties of a right triangle and standard trigonometric ratios. Given one acute angle in a right triangle, you can determine the other acute angle and then compute the cotangent of that angle using known values from the 30°–60°–90° triangle. It is a straightforward but important concept in trigonometry and geometry.

Given Data / Assumptions:

• Triangle PQR is right angled at Q, so m∠Q = 90°.
• m∠R = 30°.
• The sum of interior angles of a triangle is 180°.
• Cotangent is defined as cot θ = cos θ / sin θ or adjacent/opposite in a right triangle.

Concept / Approach: First, use the angle sum property of a triangle: the three interior angles add to 180°. With m∠Q = 90° and m∠R = 30°, we can find m∠P. Once we know that P corresponds to an angle of 60°, we recall the exact trigonometric values for a 30°–60°–90° triangle. The cotangent of 60° is the reciprocal of tan 60°, and standard tables give tan 60° = √3, so cot 60° = 1/√3.

Step-by-Step Solution: m∠P + m∠Q + m∠R = 180° for any triangle. Given m∠Q = 90° and m∠R = 30°. So m∠P = 180° − 90° − 30° = 60°. We need cot P = cot 60°. Recall that tan 60° = √3. Hence cot 60° = 1 / tan 60° = 1/√3. Therefore cot P = 1/√3.

Verification / Alternative check: In a 30°–60°–90° triangle, side lengths are in the ratio 1 : √3 : 2, with the smallest side opposite 30°, the largest side opposite 90° and the medium side opposite 60°. For angle P = 60°, cot P = adjacent/opposite = (side opposite 30°)/(side opposite 60°) = 1/√3, confirming the value obtained from the tan-cot relationship.

Why Other Options Are Wrong: 1/2 and 1/√2 are values associated with cos 60° or sin 45°, not with cot 60°. The value 2 is cot 30°, not cot 60°. √3 is tan 60°, not its reciprocal. Only 1/√3 matches the standard exact value of cot 60° and therefore cot P in this triangle.

Common Pitfalls: A common mistake is to assign 30° to P and 60° to R, reversing the angles, which leads to choosing cot 30° = √3 instead of cot 60°. Another error is mixing up tan and cot values for these standard angles. Carefully using the angle sum rule and recalling the exact trigonometric ratios prevents these errors.

Final Answer: Thus, cot P in the given triangle is 1/√3.