Use the given information to find each of the following.
sin y, given cos 2y = -1/3 , π/2 < y < π

검증된 단계별 안내

Recall the double-angle identity for cosine: \(\cos(2y) = 2\cos^2(y) - 1\).

Use the given value \(\cos(2y) = -\frac{1}{3}\) and substitute it into the identity: \(-\frac{1}{3} = 2\cos^2(y) - 1\).

Solve the equation for \(\cos^2(y)\): add 1 to both sides to get \(\frac{2}{3} = 2\cos^2(y)\), then divide both sides by 2 to find \(\cos^2(y) = \frac{1}{3}\).

Find \(\cos(y)\) by taking the square root: \(\cos(y) = \pm \sqrt{\frac{1}{3}} = \pm \frac{1}{\sqrt{3}}\). Determine the correct sign of \(\cos(y)\) using the interval \(\frac{\pi}{2} < y < \pi\), where cosine is negative.

Use the Pythagorean identity \(\sin^2(y) + \cos^2(y) = 1\) to find \(\sin(y)\): substitute \(\cos^2(y) = \frac{1}{3}\), then solve for \(\sin(y)\), considering the sign of \(\sin(y)\) in the given interval.

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주요 개념

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Double-Angle Identity for Cosine

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

Sign of Trigonometric Functions in Quadrants

The value of sin(y) depends on the quadrant where angle y lies. Since π/2 < y < π, y is in the second quadrant where sine is positive and cosine is negative. This information helps determine the correct sign of sin(y) after calculation.

Pythagorean Identity

The Pythagorean identity, sin²(y) + cos²(y) = 1, relates sine and cosine of the same angle. It is useful for finding sin(y) once cos(y) is determined or vice versa, ensuring the values satisfy this fundamental trigonometric relationship.

Use the given information to find each of the following.sin y, given cos 2y = -1/3 , π/2 < y < π