Evaluate the trigonometric expression cosec(-150°) using the standard definitions of cosecant and the sign rules for negative angles. Choose the correct numerical value.

Introduction / Context:
This question tests understanding of basic trigonometric function values at standard angles and how the functions behave for negative angles. Cosecant is the reciprocal of sine, so knowing the sine value at 150 degrees allows us to find cosec 150 degrees and then use the symmetry rule for negative angles to find cosec(-150°).

Given Data / Assumptions:

• We are asked to find cosec(-150°).
• Cosecant is defined as cosec θ = 1 / sin θ, for angles where sin θ is not zero.
• We use the standard trigonometric identities for reference angles and signs.

Concept / Approach:
First, recall that for any angle θ, sin(-θ) = -sin θ. Therefore, cosec(-θ) = 1 / sin(-θ) = 1 / (-sin θ) = -cosec θ. Next, we evaluate sin 150 degrees using the reference angle 30 degrees in the second quadrant. Once we know sin 150 degrees, we can find cosec 150 degrees and then apply the sign rule for negative angles to obtain cosec(-150°).

Step-by-Step Solution:
Use the identity sin(-θ) = -sin θ. Therefore, cosec(-θ) = -cosec θ.Compute sin 150°. The angle 150 degrees lies in the second quadrant, where sine is positive.The reference angle for 150 degrees is 30 degrees, and sin 30° = 1/2. Hence sin 150° = 1/2.Therefore, cosec 150° = 1 / sin 150° = 1 / (1/2) = 2.Using the sign rule, cosec(-150°) = -cosec 150° = -2.

Verification / Alternative check:
Another way to think about the sign is to consider the unit circle. The point corresponding to 150 degrees has a positive y coordinate of 1/2. Reflecting this point across the x axis to get -150 degrees reverses the sign of the y coordinate, so sin(-150°) = -1/2. The reciprocal of -1/2 is -2. This matches the value obtained from the algebraic identity approach, confirming that cosec(-150°) equals -2.

Why Other Options Are Wrong:

• 2/sqrt(3) and -2/sqrt(3) are related to sin 60 degrees and sin 120 degrees, not to sin 150 degrees.
• 2 is the value of cosec(150°), not cosec(-150°), so it has the wrong sign.
• 1 is not the reciprocal of sin 150 degrees and does not match any relevant standard value here.

Common Pitfalls:

• Forgetting that sine is positive in the second quadrant and mixing up the standard angle values.
• Not applying the negative angle identity correctly and assuming cosec(-θ) equals cosec θ.
• Confusing reference angles for 150 degrees with those for 120 degrees or 60 degrees.

Final Answer:
-2