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The cosine function, often denoted as \(\cos\), is a fundamental trigonometric function because it helps us understand the relationship between an angle and the adjacent side of a right triangle over the hypotenuse. In the unit circle, where the radius is one, the cosine of an angle is the horizontal coordinate of the corresponding point on the circle.In terms of quadrants:- **First Quadrant (0° to 90°):** \(\cos\) is positive.- **Second Quadrant (90° to 180°):** \(\cos\) is negative.- **Third Quadrant (180° to 270°):** \(\cos\) is negative.- **Fourth Quadrant (270° to 360°):** \(\cos\) is positive.Knowing where cosine is positive or negative helps to identify where certain conditions fit in trigonometric problems. Here, since \(\cos t < 0\), we focus on the second and third quadrants.

The cotangent function, represented as \(\cot\), is the reciprocal of the tangent function. It is essential in trigonometry for analyzing angles and understanding relationships in triangles and circles. The formula can be defined as \( \cot t = \frac{1}{\tan t} = \frac{\cos t}{\sin t} \).Understanding its signs is crucial:- **First Quadrant:** \(\cot\) is positive.- **Second Quadrant:** \(\cot\) is negative.- **Third Quadrant:** \(\cot\) is positive.- **Fourth Quadrant:** \(\cot\) is negative.In the exercise, \(\cot t < 0\) indicates a negative cotangent, which occurs in the second and fourth quadrants due to the nature of \(\tan\). This is because tangent is positive where both sine and cosine are either both positive or both negative, and conversely cotangent follows this.

Signs of trigonometric functions are essential to determining the correct quadrants in which an angle or terminal side of an angle might lie. The unit circle is divided into four quadrants, each with unique sign patterns for sine, cosine, tangent, and their reciprocals.Here's a quick recap:- **First Quadrant:** All trigonometric functions are positive.- **Second Quadrant:** \(\sin\) is positive, \(\cos\) and \(\tan\) are negative.- **Third Quadrant:** \(\tan\) is positive while \(\sin\) and \(\cos\) are negative.- **Fourth Quadrant:** \(\cos\) is positive and \(\sin\) and \(\tan\) are negative.For the given exercise, combining both conditions \(\cos t < 0\) and \(\cot t < 0\), helps narrow down the quadrant. Both conditions are satisfied in the second quadrant, where cosine is negative and cotangent is negative, allowing one to pinpoint the terminal point determined by \(t\). Recognizing these patterns is an invaluable tool in solving trigonometric problems.