Use the given information to find each of the following.
sin 2x, given sin x = 0.6, π/2 < y < π

검증된 단계별 안내

Identify the given information: \( \sin x = 0.6 \) and the angle \( x \) lies in the interval \( \frac{\pi}{2} < x < \pi \), which means \( x \) is in the second quadrant.

Recall the double-angle formula for sine: \( \sin 2x = 2 \sin x \cos x \). To find \( \sin 2x \), we need both \( \sin x \) and \( \cos x \).

Use the Pythagorean identity to find \( \cos x \): \( \cos x = \pm \sqrt{1 - \sin^2 x} \). Since \( x \) is in the second quadrant, where cosine is negative, choose the negative root.

Calculate \( \cos x = -\sqrt{1 - (0.6)^2} \) without simplifying the square root fully, just set up the expression.

Substitute \( \sin x = 0.6 \) and the expression for \( \cos x \) into the double-angle formula \( \sin 2x = 2 \times 0.6 \times \cos x \) to express \( \sin 2x \) in terms of known values.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Double-Angle Identity for Sine

The double-angle identity for sine states that sin(2x) = 2 sin(x) cos(x). This formula allows you to find the sine of twice an angle if you know the sine and cosine of the original angle.

Using the Pythagorean Identity to Find Cosine

Given sin(x), you can find cos(x) using the Pythagorean identity: sin²(x) + cos²(x) = 1. Rearranging gives cos(x) = ±√(1 - sin²(x)). The sign depends on the quadrant where x lies.

Determining the Sign of Trigonometric Functions Based on Quadrants

The sign of sine and cosine depends on the quadrant of the angle. Since π/2 < x < π (second quadrant), sin(x) is positive and cos(x) is negative. This helps determine the correct sign for cos(x) when using the Pythagorean identity.

Use the given information to find each of the following.sin 2x, given sin x = 0.6, π/2 < y < π