Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. √3cot θ = - —— 3

Verified step by step guidance

1. Recognize that \( \cot \theta = \frac{1}{\tan \theta} \), so \( \tan \theta = -\frac{3}{\sqrt{3}} \).
2. Simplify \( \tan \theta = -\frac{3}{\sqrt{3}} \) to \( \tan \theta = -\sqrt{3} \).
3. Recall that \( \tan \theta = -\sqrt{3} \) corresponds to angles where the tangent function is negative, which occurs in the second and fourth quadrants.
4. Identify the reference angle for \( \tan \theta = \sqrt{3} \), which is \( 60^\circ \) or \( \frac{\pi}{3} \) radians.
5. Determine the angles in the specified interval [0°, 360°) that have a tangent of \(-\sqrt{3}\), which are \( 120^\circ \) and \( 300^\circ \).

Key Concepts

Cotangent Function

The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function. It is defined as cot(θ) = cos(θ)/sin(θ). Understanding cotangent is essential for solving the equation provided, as it relates the angle θ to the ratio of the adjacent side to the opposite side in a right triangle.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to specific signs of the sine and cosine functions. In the context of cotangent, knowing which quadrants yield negative values is crucial. Since cot(θ) is negative in the second and fourth quadrants, this information helps narrow down the possible angles for θ.

Reference Angles

A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. It is used to find the values of trigonometric functions in different quadrants. For the given cotangent value, determining the reference angle will allow us to find all corresponding angles in the specified interval [0°, 360°).