Find sinθ.
cot θ = -1/3, θ in quadrant IV

Verified step by step guidance

1. Recall the definition of cotangent in terms of sine and cosine: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).
2. Given \(\cot \theta = -\frac{1}{3}\), set \(\frac{\cos \theta}{\sin \theta} = -\frac{1}{3}\), which implies \(\cos \theta = -\frac{1}{3} \sin \theta\).
3. Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) and substitute \(\cos \theta\) from the previous step: \(\sin^2 \theta + \left(-\frac{1}{3} \sin \theta\right)^2 = 1\).
4. Simplify the equation to find \(\sin^2 \theta\): \(\sin^2 \theta + \frac{1}{9} \sin^2 \theta = 1\), which combines to \(\frac{10}{9} \sin^2 \theta = 1\).
5. Solve for \(\sin \theta\) by isolating it and taking the square root, then determine the correct sign of \(\sin \theta\) based on the fact that \(\theta\) is in quadrant IV, where sine is negative.

Key Concepts

Cotangent and its Relationship to Sine and Cosine

Cotangent (cot θ) is the ratio of the adjacent side to the opposite side in a right triangle, or equivalently, cot θ = cos θ / sin θ. Knowing cot θ allows us to express sine and cosine in terms of each other, which is essential for finding sin θ when cot θ is given.

Sign of Trigonometric Functions in Quadrants

The sign of sine and cosine depends on the quadrant where the angle θ lies. In quadrant IV, sine is negative and cosine is positive. This information helps determine the correct sign of sin θ after calculating its magnitude.

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1. This relationship allows us to find one trigonometric function if the other is known, which is useful when cot θ is given and we need to find sin θ.