Use the given information to find each of the following.
sin x, given cos 2x = 2/3 , with π < x < 3π/2

Verified step by step guidance

1. Start by using the double angle identity for cosine: \( \cos 2x = 1 - 2\sin^2 x \).
2. Substitute the given value into the identity: \( \frac{2}{3} = 1 - 2\sin^2 x \).
3. Rearrange the equation to solve for \( \sin^2 x \): \( 2\sin^2 x = 1 - \frac{2}{3} \).
4. Simplify the equation: \( 2\sin^2 x = \frac{1}{3} \), then \( \sin^2 x = \frac{1}{6} \).
5. Since \( \pi < x < \frac{3\pi}{2} \), \( \sin x \) is negative in this quadrant, so \( \sin x = -\sqrt{\frac{1}{6}} \).

Key Concepts

Double Angle Formula for Cosine

The double angle formula for cosine states that cos(2x) can be expressed in terms of sin(x) and cos(x) as cos(2x) = cos²(x) - sin²(x) or alternatively as cos(2x) = 2cos²(x) - 1 or cos(2x) = 1 - 2sin²(x). This formula is essential for relating the cosine of a double angle to the sine and cosine of the original angle, which is crucial for solving the given problem.

Sine and Cosine Relationship

The relationship between sine and cosine is fundamental in trigonometry, expressed as sin²(x) + cos²(x) = 1. This identity allows us to find one trigonometric function if we know the other. In this problem, knowing cos(2x) enables us to derive sin(x) using this relationship, especially since we can express cos(2x) in terms of sin(x).

Quadrants of the Unit Circle

Understanding the unit circle and the corresponding quadrants is vital for determining the signs of sine and cosine values. In this case, the interval π < x < 3π/2 indicates that x is in the third quadrant, where sine is negative and cosine is also negative. This knowledge helps in accurately determining the value of sin(x) after calculating it from cos(2x).