Problem 52 - Algebra and Trigonometry

Question

From the information given, find the quadrant in which the terminal point determined by \(t\) lies. $$ \cos t<0 \text { and } \cot t<0 $$

Step-by-Step Solution

Answer

The terminal point lies in the second quadrant.

Step 1: Understanding the Problem

We are given that \( \cos t < 0 \) and \( \cot t < 0 \). We need to determine in which quadrant this terminal point lies. The cosine function is negative in the second and third quadrants. The cotangent function, \( \cot t = \frac{\cos t}{\sin t} \), is negative when cosine and sine have opposite signs.

Step 2: Analyzing Cosine Function

Since \( \cos t < 0 \), we know \( t \) could be in the second or third quadrant. In these quadrants, cosine is negative. We will use this information paired with the behavior of \( \cot t \) to narrow down further.

Step 3: Analyzing Cotangent Function

Since \( \cot t < 0 \), cosine and sine must have opposite signs. In the second quadrant, cosine is negative and sine is positive, yielding a negative cotangent. In the third quadrant, both sine and cosine are negative, yielding a positive cotangent.

Step 4: Combining Both Conditions

Given the above analysis: \( \cos t < 0 \) applies to the second and third quadrants. \( \cot t < 0 \) only applies to the second quadrant because cosine and sine have opposite signs. Thus, the terminal point determined by \( t \) lies in the second quadrant.

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions, often abbreviated as "cos". It is related to the coordinates of points on a unit circle. When the terminal side of an angle lies in a particular quadrant, the sign of the cosine function provides essential clues about the angle's exact quadrant location.

• In the first quadrant, all trigonometric functions, including cosine, are positive.
• In the second quadrant, cosine is negative, while sine is positive.
• The third quadrant has both cosine and sine as negative.
• Finally, in the fourth quadrant, cosine is positive, and sine is negative.

Understanding this helps in problems like the given exercise, where determining the sign of cosine can immediately narrow down which quadrant the angle resides in.

Cotangent Function

The cotangent function, abbreviated as "cot", is defined as the reciprocal of the tangent function — \( \cot t = \frac{1}{\tan t} = \frac{\cos t}{\sin t} \). Its sign depends on both the cosine and sine functions as it is their ratio. The cotangent function provides insights into the angle's quadrant by analyzing the behavior of sine and cosine.

• In the first quadrant, \( \cot t \) is positive, since both sine and cosine are positive.
• In the second quadrant, \( \cot t \) becomes negative because sine is positive and cosine is negative.
• In the third quadrant, where both cosine and sine are negative, \( \cot t \) is positive.
• In the fourth quadrant, \( \cot t \) is negative, since cosine is positive and sine is negative.

In the exercise, the angle \( t \) is in the second quadrant where \( \cot t \) is negative due to cosine being negative and sine positive.

Trigonometric Functions

Trigonometric functions are fundamental in understanding angles and their relationships in a circle. These functions include sine, cosine, and tangent, among others, and are defined based on the unit circle. Each trigonometric function has a unique way of indicating an angle's position in one of the four quadrants of a circle.

• The sine function measures the y-coordinate of the corresponding point on the unit circle.
• Cosine measures the x-coordinate.
• Tangent is the ratio of sine to cosine — \( \tan t = \frac{\sin t}{\cos t} \).

Each trigonometric function changes sign depending on the quadrant, providing useful information about angles. In the given exercise, both the cosine and cotangent functions are negative, helping us pinpoint that the angle \( t \) lies in the second quadrant, where cosine is negative while cotangent captures the opposite signs of sine and cosine.