Math Problem Statement

Find the sine of the angle in radians, -11π/6.

Solution

The question in the image asks you to find the sine of the angle given in radians, 11π6-\frac{11\pi}{6}.

To solve this:

  1. Identify the reference angle:
    11π6-\frac{11\pi}{6} is in the fourth quadrant because when you add 2π2\pi (full circle) to it, you get: 11π6+2π=11π6+12π6=π6-\frac{11\pi}{6} + 2\pi = -\frac{11\pi}{6} + \frac{12\pi}{6} = \frac{\pi}{6} This is equivalent to 11π6\frac{11\pi}{6}, but the angle is in the fourth quadrant.

  2. Determine the sine of the reference angle:
    The reference angle for π6\frac{\pi}{6} is π6\frac{\pi}{6} itself.

    The sine of π6\frac{\pi}{6} is: sin(π6)=12\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}

  3. Adjust for the quadrant:
    Since the angle is in the fourth quadrant, the sine value is negative.

    Therefore: sin(11π6)=12\sin\left(-\frac{11\pi}{6}\right) = -\frac{1}{2}

The correct answer is 12\boxed{-\frac{1}{2}}.

Would you like further details or have any other questions?

Here are 5 related questions to explore further:

  1. What is the cosine of 11π6-\frac{11\pi}{6}?
  2. How would you find the tangent of 7π4\frac{7\pi}{4}?
  3. What is the reference angle for an angle in the third quadrant?
  4. How does the sine function behave for angles in the second quadrant?
  5. Can you describe the process of converting an angle from degrees to radians?

Tip: When working with trigonometric functions, always consider the angle's quadrant to determine the sign of the result.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Unit Circle
Reference Angles

Formulas

sin(θ) = sin(reference angle)
Reference Angle Calculation
Sine in Fourth Quadrant

Theorems

Unit Circle Theorem
Reference Angle Theorem

Suitable Grade Level

Grades 10-12