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

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Recall the double-angle identity for cosine: \(\cos 2x = 1 - 2\sin^2 x\). This relates \(\cos 2x\) to \(\sin x\).

Substitute the given value \(\cos 2x = \frac{2}{3}\) into the identity: \(\frac{2}{3} = 1 - 2\sin^2 x\).

Rearrange the equation to solve for \(\sin^2 x\): \(2\sin^2 x = 1 - \frac{2}{3}\), then simplify the right side.

Calculate \(\sin^2 x\) by dividing both sides by 2: \(\sin^2 x = \frac{1 - \frac{2}{3}}{2}\).

Determine the sign of \(\sin x\) based on the interval \(\pi < x < \frac{3\pi}{2}\). Since \(x\) is in the third quadrant, where sine is negative, take the negative square root to find \(\sin x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Double-Angle Identity for Cosine

The double-angle identity states that cos(2x) = 2cos²(x) - 1 or cos(2x) = 1 - 2sin²(x). This identity allows us to express cos(2x) in terms of sin(x) or cos(x), which is essential for finding sin(x) when cos(2x) is known.

Determining the Sign of Trigonometric Functions Based on Quadrants

The value of sin(x) and cos(x) depends on the quadrant where angle x lies. Since π < x < 3π/2 places x in the third quadrant, both sine and cosine values are negative there. This information helps determine the correct sign of sin(x) after calculation.

Using Pythagorean Identities to Relate Sine and Cosine

The Pythagorean identity sin²(x) + cos²(x) = 1 links sine and cosine values. After expressing cos(2x) in terms of sin(x) or cos(x), this identity helps solve for sin(x) by substituting and rearranging the equation.

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