In trigonometry using radian measure, what is the exact value of cosec(3π/4), the cosecant of three pi by four radians?

Introduction / Context:
This question tests your understanding of standard trigonometric values for special angles, here expressed in radians. Being able to convert angles between degrees and radians and recall exact values for sine, cosine, and related functions such as cosecant is very important in both school mathematics and many competitive exams.

Given Data / Assumptions:

• The angle is 3π/4 radians.
• We are asked to find cosec(3π/4).
• cosec x is defined as 1 / sin x.
• We work with standard values on the unit circle.

Concept / Approach:
Key ideas:

• 3π/4 radians corresponds to 135 degrees.
• The sine of 135 degrees (or 3π/4) is sin(180° − 45°) which equals sin 45°.
• sin 45° = √2 / 2.
• cosec x is the reciprocal of sin x, so cosec x = 1 / sin x.

Step-by-Step Solution:
Convert 3π/4 radians to degrees: (3π/4) * (180° / π) = 3 * 45° = 135°. Recognise that 135° = 180° − 45° and lies in the second quadrant. In the second quadrant, sine is positive, so sin 135° = sin 45°. Use the standard value sin 45° = √2 / 2. Therefore, sin(3π/4) = √2 / 2. Compute cosec(3π/4) = 1 / sin(3π/4) = 1 / (√2 / 2) = 2 / √2 = √2.

Verification / Alternative check:
Using the unit circle, the point corresponding to 135° has coordinates (−√2 / 2, √2 / 2). The y coordinate is the sine value, which is √2 / 2. Taking its reciprocal again gives √2. This agrees with the analytic computation and confirms the answer.

Why Other Options Are Wrong:
Option a: −√2 would be the cosecant if sine were negative, but in the second quadrant sine is positive.
Option c: 2 / √3 is associated with angles like 60°, not 135°.
Option d: −2 / √3 is incorrect for the same reason and also has the wrong sign.
Option e: −1 / √2 is simply the negative of sin 45°, not its reciprocal, and has an incorrect sign.

Common Pitfalls:
Students often forget the sign of trigonometric functions in different quadrants or mix up reciprocal values. Confusing 135° with 45° or 225° can also lead to errors. Always check the quadrant and remember that cosecant is the reciprocal of sine, not cosine or tangent.

Final Answer:
The exact value of cosec(3π/4) is √2.