Find the degree measure of θ if it exists. Do not use a calculator.
θ = cot⁻¹ (-√3/3)

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Recall that the inverse cotangent function, \(\cot^{-1}(x)\), gives an angle \(\theta\) such that \(\cot(\theta) = x\). Here, we want to find \(\theta\) where \(\cot(\theta) = -\frac{\sqrt{3}}{3}\).

Recognize the positive value \(\frac{\sqrt{3}}{3}\) as a common cotangent value. Since \(\cot(\theta) = \frac{1}{\tan(\theta)}\), find the angle whose tangent is the reciprocal: \(\tan(\theta) = \sqrt{3}\) corresponds to \(\theta = 60^\circ\).

Since the cotangent value is negative, determine in which quadrants cotangent is negative. Cotangent is negative where tangent is negative, which occurs in the second and fourth quadrants.

Use the reference angle \(60^\circ\) and find the angles in the second and fourth quadrants: \(180^\circ - 60^\circ = 120^\circ\) and \(360^\circ - 60^\circ = 300^\circ\).

Recall the principal value range for \(\cot^{-1}(x)\) is usually \(0^\circ < \theta < 180^\circ\). Therefore, select the angle within this range that satisfies \(\cot(\theta) = -\frac{\sqrt{3}}{3}\), which is \(120^\circ\).

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Inverse Cotangent Function (cot⁻¹)

The inverse cotangent function, cot⁻¹, returns the angle whose cotangent is a given value. It is the inverse of the cotangent function, which relates an angle to the ratio of the adjacent side over the opposite side in a right triangle. Understanding its range and output values is essential for finding θ.

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Introduction to Cotangent Graph

Cotangent Values and Special Angles

Cotangent values for special angles like 30°, 45°, and 60° are commonly memorized. For example, cot 30° = √3, cot 45° = 1, and cot 60° = √3/3. Recognizing these values helps identify the angle corresponding to a given cotangent without a calculator.

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Sign and Quadrant Considerations

The sign of the cotangent value indicates the quadrant where the angle lies. Since cotangent is negative, θ must be in a quadrant where cotangent is negative (second or fourth quadrant). Knowing the principal range of cot⁻¹ and quadrant signs helps determine the correct angle measure.

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