What is the exact value of cosec(−7π/6) when the angle is measured in radians on the unit circle?

Introduction / Context:This question evaluates your understanding of trigonometric functions for angles expressed in radians and the behaviour of sine and cosecant under sign changes. To find cosec(−7π/6), you must first compute sin(−7π/6) using unit circle knowledge and symmetry, and then take its reciprocal. Recognising equivalent angles and quadrants is crucial for getting the sign correct.

Given Data / Assumptions:

• The angle is −7π/6 radians.
• cosec θ is defined as 1 / sin θ.
• We use exact trigonometric values for special angles like 30° and 150°.
• We assume the reader can convert between radians and degrees if needed.

Concept / Approach:First, interpret −7π/6 in degrees to identify the reference angle and quadrant. Since π radians equals 180°, −7π/6 corresponds to −210°. Using the identity sin(−θ) = −sin θ, we reduce the problem to finding sin 210° and then adjusting the sign. Alternatively, we can add 2π to −7π/6 to get a coterminal angle in the interval [0, 2π). Once sin(−7π/6) is found, cosec(−7π/6) is simply its reciprocal.

Step-by-Step Solution:Convert −7π/6 to degrees: −7π/6 * (180/π) = −210°.Use the identity sin(−θ) = −sin θ: sin(−210°) = −sin 210°.Recognise that 210° = 180° + 30°, an angle in the third quadrant where sine is negative.Therefore sin 210° = −sin 30° = −1/2.So sin(−210°) = −(−1/2) = 1/2.Now compute cosec(−7π/6) = 1 / sin(−7π/6) = 1 / (1/2) = 2.

Verification / Alternative check:Instead of converting to degrees, we can work entirely in radians. Note that −7π/6 + 2π = −7π/6 + 12π/6 = 5π/6, which is coterminal with −7π/6. Hence sin(−7π/6) = sin(5π/6). Since 5π/6 corresponds to 150°, which lies in the second quadrant, sin 5π/6 = sin 150° = 1/2. Thus cosec(−7π/6) = cosec(5π/6) = 1 / (1/2) = 2, consistent with our previous calculation.

Why Other Options Are Wrong:

• −2 would be the cosecant if the sine value were −1/2, but we have sin(−7π/6) = +1/2.
• 2/√3 and −2/√3 correspond to angles where the sine is ±√3/2, not ±1/2.
• 0 is impossible for cosecant because it is the reciprocal of sine, and sine never equals infinity; cosecant is undefined where sine is zero.

Common Pitfalls:

• Incorrectly identifying the quadrant of the angle or the sign of sine.
• Forgetting that adding 2π or 360° to an angle yields a coterminal angle with the same sine value.
• Misapplying the identity sin(−θ) = −sin θ and reversing the sign twice.

Final Answer:2