Difficulty: Easy
Correct Answer: -1/2
Explanation:
Introduction / Context:
This problem tests your familiarity with sine values at standard angles on the unit circle. Knowing how to evaluate sine at common radian measures is very useful in trigonometry and aptitude tests.
Given Data / Assumptions:
Concept / Approach:
The angle 11π/6 corresponds to 330 degrees. This is 360 degrees minus 30 degrees, so the reference angle is 30 degrees. In the fourth quadrant, the sine function is negative, while cosine is positive. Therefore, sin(11π/6) will have the same magnitude as sin 30 degrees but with a negative sign.
Step-by-Step Solution:
Step 1: Convert 11π/6 to degrees: 11π/6 * 180/π = 11 * 30 = 330 degrees.Step 2: Recognize that 330 degrees is 360 degrees − 30 degrees, so the reference angle is 30 degrees.Step 3: Recall that sin 30 degrees = 1/2.Step 4: Since 330 degrees lies in the fourth quadrant, where sine is negative, sin(330°) = −sin 30°.Step 5: Therefore sin(11π/6) = −1/2.
Verification / Alternative check:
You can think of the point on the unit circle for 330 degrees. The coordinates are (cos 330°, sin 330°) = (√3/2, −1/2). The sine coordinate is clearly −1/2, which matches the result obtained by using the reference angle method.
Why Other Options Are Wrong:
- 2/√3 and −2/√3 correspond to cosine or sine values at 60 degrees or related special angles, not at 330 degrees.
- 1/2 is the sine of 30 degrees and would be correct for a first quadrant or reference angle without considering the sign change in the fourth quadrant.
- √3/2 does not have the correct magnitude for sine at 330 degrees; it corresponds instead to cosine at 30 degrees.
Common Pitfalls:
Students often forget to apply the correct sign of the trigonometric function in each quadrant and may simply use the reference angle value. Always remember that sine is negative in the third and fourth quadrants. Using a quick sign diagram for quadrants helps avoid this error.
Final Answer:
The value of sin(11π/6) is −1/2.
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