91Ó°ÊÓ

When it comes to angles, degrees are perhaps the most familiar unit; however, in trigonometry, radians are the preferred measure for angle calculations. A radian is based on the radius of a circle. Specifically, one radian is the angle made when the arc length is equal to the radius of the circle. This relationship is key because it ties directly into the geometry of a circle, making calculations more natural in mathematical analysis.
In the solution, when angles appear as multiples of \( \pi \), they're measured in radians. For instance, \( \pi \) radians is equivalent to 180 degrees, forming a direct link between the two systems. To find solutions in the least nonnegative angle, we often work within the range from 0 to \( 2\pi \). This range describes a full circle, where starting angles reset after \( 2\pi \) radians.
Radians are effective as they allow for more streamlined calculus operations and are essential when working with periodic functions like sine, cosine, and tangent seen in trigonometry.

A reference angle is a simplified version of an angle, stripped of its direction in a unit circle, making calculations easier. The reference angle of any angle is essentially the smallest angle between the terminal side of the original angle and the horizontal axis. When it comes to solving trigonometric equations, the reference angle helps determine where the given angle's trigonometric function might take on positive or negative values.
Consider the function \( \tan x = -\frac{1}{\sqrt{3}} \). The equation implies that the tangent corresponds to specific angles where the tangent value would be \( \frac{1}{\sqrt{3}} \) referenced to either the second or fourth quadrant, where the tangent is negative. Hence, the reference angle of \( \frac{\pi}{6} \) lets us identify the key angles \( \frac{5\pi}{6} \) and \( \frac{11\pi}{6} \), which lie in these specific quadrants.

The cotangent function, denoted as \( \cot x \), is the reciprocal of the tangent function. Simply put, if the tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle, the cotangent is the ratio of the adjacent side to the opposite side.
Mathematically, this means \( \cot x = \frac{1}{\tan x} \). This equality is pivotal in transforming a problem involving cotangent to one involving tangent, which can be more straightforward to handle, especially when employing angle measures like radians.
For the equation \( \cot x = -\sqrt{3} \), it is equivalent to \( \tan x = -\frac{1}{\sqrt{3}} \), thereby enabling us to consider the angles at which this specific tangent value as achieved. Recognizing these transformations enhances the ease of solving trigonometric equations.

The tangent function, \( \tan x \), is a fundamental trigonometric function. It expresses the ratio of the opposite side to the adjacent side in a right-angled triangle. Unlike sine and cosine functions, the tangent function can take any real number value, ranging from negative infinity to positive infinity.
In solving equations like \( \tan x = -\frac{1}{\sqrt{3}} \), understanding the behavior of the tangent function is crucial. The tangent function is periodic with a period of \( \pi \), meaning its values repeat every \( \pi \) radians. This periodicity helps in finding all possible solutions to the equation within a given range, like \([0, 2\pi) \).
Knowing that the function is negative in the second and fourth quadrants provides insight into solving for the angles. Thus, the respective angles are \( \frac{5\pi}{6} \) and \( \frac{11\pi}{6} \), using the reference angle \( \frac{\pi}{6} \) to establish these angles' exact positions.