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

Verified step by step guidance

1. Recall the Pythagorean identity: \(\sin^{2} \theta + \cos^{2} \theta = 1\).
2. Substitute the given value of \(\cos \theta = \frac{5}{6}\) into the identity: \(\sin^{2} \theta + \left(\frac{5}{6}\right)^{2} = 1\).
3. Calculate \(\left(\frac{5}{6}\right)^{2}\) which is \(\frac{25}{36}\), so the equation becomes \(\sin^{2} \theta + \frac{25}{36} = 1\).
4. Isolate \(\sin^{2} \theta\) by subtracting \(\frac{25}{36}\) from both sides: \(\sin^{2} \theta = 1 - \frac{25}{36}\).
5. Since \(\theta\) is in quadrant I, where sine is positive, take the positive square root: \(\sin \theta = \sqrt{1 - \left(\frac{5}{6}\right)^{2}}\).

Key Concepts

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.