Find sin θ.
cos θ = 5/6, θ in quadrant I

검증된 단계별 안내

Recall the Pythagorean identity: \(\sin^{2} \theta + \cos^{2} \theta = 1\).

Substitute the given value of \(\cos \theta = \frac{5}{6}\) into the identity: \(\sin^{2} \theta + \left(\frac{5}{6}\right)^{2} = 1\).

Calculate \(\left(\frac{5}{6}\right)^{2}\) which is \(\frac{25}{36}\), so the equation becomes \(\sin^{2} \theta + \frac{25}{36} = 1\).

Isolate \(\sin^{2} \theta\) by subtracting \(\frac{25}{36}\) from both sides: \(\sin^{2} \theta = 1 - \frac{25}{36}\).

Since \(\theta\) is in quadrant I, where sine is positive, take the positive square root: \(\sin \theta = \sqrt{1 - \left(\frac{5}{6}\right)^{2}}\).

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주요 개념

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

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. This relationship allows you to find the sine of an angle if the cosine is known, by rearranging the formula to sin θ = ±√(1 - cos²θ).

Sign of Trigonometric Functions in Quadrants

The sign of sine and cosine depends on the quadrant where the angle lies. In quadrant I, both sine and cosine values are positive, which helps determine the correct sign when calculating sin θ from the Pythagorean identity.

Using Given Cosine Value to Find Sine

Given cos θ = 5/6, you substitute this value into the Pythagorean identity to find sin θ. Since θ is in quadrant I, sin θ will be positive, so sin θ = √(1 - (5/6)²) = √(1 - 25/36) = √(11/36) = √11/6.

Find sin θ.cos θ = 5/6, θ in quadrant I