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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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.

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Double Angle Identities

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.

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Introduction to Trigonometric Functions

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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