Find the exact value of cosec 120° using standard trigonometric ratios and quadrant rules.

Introduction / Context:
This question tests your understanding of trigonometric values at special angles and your knowledge of signs in different quadrants. Cosecant is the reciprocal of sine, so evaluating cosec 120 degrees requires you to know sin 120 degrees exactly. These exact trigonometric values are fundamental and appear frequently in both school mathematics and aptitude exams.

Given Data / Assumptions:
- The angle is 120 degrees.
- We must find cosec 120°, which is 1 / sin 120°.
- All calculations should be in exact form, not decimals.

Concept / Approach:
Angles between 90 and 180 degrees lie in the second quadrant, where sine is positive and cosine is negative. The reference angle for 120 degrees is 60 degrees, so sin 120 degrees equals sin 60 degrees but with the sign appropriate to the second quadrant. We then take the reciprocal of sine to obtain cosecant.

Step-by-Step Solution:
Step 1: Recognise that 120 degrees lies in the second quadrant, between 90 and 180 degrees. Step 2: Find the reference angle: 180 degrees − 120 degrees = 60 degrees, so the reference angle is 60 degrees. Step 3: Recall that sin 60 degrees = √3 / 2. Step 4: In the second quadrant, sine is positive, so sin 120 degrees = sin 60 degrees = √3 / 2. Step 5: Cosecant is the reciprocal of sine, so cosec 120 degrees = 1 / sin 120 degrees = 1 / (√3 / 2). Step 6: Simplify the reciprocal: 1 / (√3 / 2) = 2 / √3. Step 7: Thus, the exact value of cosec 120 degrees is 2 / √3.
Verification / Alternative check:
You can verify the sign by using the unit circle. At 120 degrees, the coordinates of the point on the unit circle have x coordinate negative and y coordinate positive, and the y coordinate is sin 120 degrees. Because the y coordinate is positive, sine must be positive, confirming that sin 120 degrees is +√3 / 2 rather than negative. Therefore, its reciprocal is also positive, which matches 2 / √3.

Why Other Options Are Wrong:
Option b, 2, would be the reciprocal of sin 30 degrees, not 120 degrees.
Option c, −2/√3, has the correct magnitude but the wrong sign; it would correspond to a third or fourth quadrant angle where sine is negative.
Option d, −2, has both incorrect magnitude and sign for this angle.
Option e, √3/2, is actually the value of sin 60 degrees or sin 120 degrees, not its reciprocal.

Common Pitfalls:
Students sometimes confuse the reference angle with the actual angle and assign the wrong sign to the sine or cosine values. Another frequent mistake is inverting the wrong function or forgetting that cosecant is the reciprocal of sine, not cosine. A further common error is to rationalise the denominator incorrectly, but here it is acceptable to leave the denominator as √3 unless a specific format is required. Keeping track of quadrants and signs systematically helps avoid these mistakes.

Final Answer:
The exact value of cosec 120° is 2 / √3.