Find sinθ.
cos (-θ) = (√3)/6, cot θ < 0

Verified step by step guidance

1. Recall the even-odd properties of trigonometric functions: since cosine is an even function, we have \(\cos(-\theta) = \cos(\theta)\). Therefore, \(\cos(\theta) = \frac{\sqrt{3}}{6}\).
2. Identify the quadrant where \(\theta\) lies using the given condition \(\cot \theta < 0\). Since \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), for \(\cot \theta\) to be negative, sine and cosine must have opposite signs.
3. Given that \(\cos \theta = \frac{\sqrt{3}}{6}\) is positive, and \(\cot \theta < 0\), it follows that \(\sin \theta\) must be negative. This places \(\theta\) in the fourth quadrant, where cosine is positive and sine is negative.
4. Use the Pythagorean identity to find \(\sin \theta\): \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\cos \theta = \frac{\sqrt{3}}{6}\) and solve for \(\sin^2 \theta\).
5. Take the square root of \(\sin^2 \theta\) to find \(\sin \theta\). Since \(\sin \theta\) is negative in the fourth quadrant, choose the negative root to determine \(\sin \theta\).

Key Concepts

Even and Odd Properties of Trigonometric Functions

Cosine is an even function, meaning cos(-θ) = cos(θ). This property allows us to replace cos(-θ) with cos(θ) when solving for θ, simplifying the problem by focusing on positive angle values.

Sign of Trigonometric Functions in Quadrants

The sign of trigonometric functions depends on the quadrant of the angle. Since cotangent is negative, θ lies in either the second or fourth quadrant, where sine and cosine have specific positive or negative values that help determine the correct angle.

Relationship Between Cotangent, Sine, and Cosine

Cotangent is the ratio of cosine to sine (cot θ = cos θ / sin θ). Knowing cot θ < 0 helps infer the signs of sine and cosine, which is essential to find the correct value of sin θ consistent with the given conditions.