What is the exact value of the trigonometric expression cosec(11π/6), where the angle is measured in radians?

Introduction / Context:
This question requires evaluation of a trigonometric function at a standard angle expressed in radians. The angle 11π/6 corresponds to 330 degrees. To find cosec(11π/6), we first find sin(11π/6) and then take its reciprocal. Understanding the unit circle and the relationship between radians and degrees is very helpful here.

Given Data / Assumptions:

• The angle is 11π/6 radians.
• We must compute the exact value of cosec(11π/6).
• Cosecant is defined as 1 / sin θ.
• We use standard exact values of sine for special angles on the unit circle.

Concept / Approach:
First convert 11π/6 to degrees to recognise the reference angle. Since π radians = 180 degrees, 11π/6 = 11 * 180 / 6 = 330 degrees. The sine of 330 degrees is negative because the angle lies in the fourth quadrant, where sine is negative. Specifically, 330 degrees has a reference angle of 30 degrees, and sin 330 degrees = −sin 30 degrees. Once sin(11π/6) is known, cosec(11π/6) is simply its reciprocal.

Step-by-Step Solution:
Convert 11π/6 to degrees: 11π/6 * (180/π) = 11 * 30 = 330 degrees.In the unit circle, sin 330 degrees = −sin 30 degrees because 330 degrees is in the fourth quadrant.Recall sin 30 degrees = 1/2, so sin 330 degrees = −1/2.Therefore, sin(11π/6) = −1/2.Cosecant is the reciprocal of sine: cosec(11π/6) = 1 / sin(11π/6) = 1 / (−1/2) = −2.

Verification / Alternative check:
Another way is to consider the unit circle coordinates. The angle 11π/6 has coordinates (cos 11π/6, sin 11π/6) = (cos 330°, sin 330°) = (√3/2, −1/2). These are standard values. Since sin 11π/6 = −1/2, its reciprocal is −2, confirming the previous calculation. The sign is negative in the fourth quadrant, where sine is negative but cosine is positive.

Why Other Options Are Wrong:

• 2 would be cosec of 30 degrees, not 330 degrees where sine is negative.
• −2/√3 and ±√2 involve sqrt(3) or sqrt(2) and correspond to different standard angles such as 60 degrees or 45 degrees.
• Only −2 matches the reciprocal of sin(11π/6) = −1/2.

Common Pitfalls:

• Forgetting that 11π/6 lies in the fourth quadrant and therefore sine is negative.
• Mixing up the sine values of 30 degrees and 60 degrees.
• Incorrectly converting from radians to degrees, which changes the angle and hence the trigonometric values.

Final Answer:
-2