In right triangle XYZ, angle Y is 90°. If ∠Z = 60°, determine the exact value of Cosec X, where X is the remaining acute angle of the triangle.

Introduction / Context:
This trigonometry question deals with a right angled triangle and asks for the cosecant of one of its acute angles, given the other acute angle. In a right triangle, once one acute angle is known, the other is determined, and special angle values can be used. This problem reinforces the relationship between complementary angles and the known trigonometric values for 30° and 60°.

Given Data / Assumptions:

• Triangle XYZ is right angled at Y, so ∠Y = 90°.
• One acute angle is ∠Z = 60°.
• The remaining acute angle is ∠X, which we must identify.
• We are asked to find Cosec X, the reciprocal of sin X.

Concept / Approach:
In any triangle, the three interior angles sum to 180°. If one angle is 90° and another is 60°, the third angle must be 30°. Once we know that ∠X = 30°, we can use the standard special angle values: sin 30° = 1/2, so Cosec 30° = 1 / sin 30° = 2. The problem does not require side lengths; it only requires understanding of angle relationships and basic trigonometric values.

Step-by-Step Solution:
1) The sum of angles in triangle XYZ is 180°. 2) We are given ∠Y = 90° and ∠Z = 60°. 3) Find ∠X: ∠X = 180° − ∠Y − ∠Z = 180° − 90° − 60° = 30°. 4) Recall the standard trigonometric value sin 30° = 1/2. 5) Cosec X is defined as 1 / sin X. Since X = 30°, we have Cosec X = 1 / sin 30°. 6) Substitute sin 30° = 1/2 to get Cosec X = 1 / (1/2) = 2.

Verification / Alternative check:
We can imagine a 30°–60°–90° triangle with side lengths in the ratio 1 : √3 : 2, where 1 is opposite 30°, √3 is opposite 60°, and 2 is the hypotenuse. At angle X = 30°, sin X is opposite / hypotenuse = 1 / 2. This confirms that sin 30° = 1/2 and therefore Cosec 30° = 2. The reasoning is consistent with standard geometry, so the answer is verified.

Why Other Options Are Wrong:
Option b (1/2) is the sine, not the cosecant, of 30°. Option c (1/√2) is associated with angles of 45°, not 30°. Option d (1/√3) is related to sin 60° or cos 30°, and option e (√3) is related to Cosec 60° or Sec 30°, not Cosec 30°. Only option a, 2, corresponds to the correct value of Cosec 30° in this triangle.

Common Pitfalls:
A common error is to confuse which angle is 30° and which is 60°, or to assume that Cosec 60° is required instead of Cosec X. Another mistake is mis remembering the basic special angle values, such as thinking sin 30° = √3 / 2 instead of 1/2. Carefully summing the angles to identify ∠X and then using the correct special angle values avoids these issues.

Final Answer:
Since ∠X = 30° in the given right triangle, the exact value of Cosec X is 2.