Problem 54 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The terminal point determined by \(t\) lies in Quadrant II.

Step 1: Understand the Quadrants

In the coordinate plane, there are four quadrants. Quadrant I (1) is where both x and y are positive. Quadrant II (2) is where x is negative and y is positive. Quadrant III (3) is where both x and y are negative. Quadrant IV (4) is where x is positive and y is negative.

Step 2: Analyze the Condition for \(\cos t < 0\)

The cosine of an angle \(t\) is negative in quadrants where the x-coordinate is negative. Therefore, \(\cos t < 0\) can only occur in Quadrant II and Quadrant III.

Step 3: Analyze the Condition for \(\cot t < 0\)

The cotangent of an angle \(t\) is defined as \(\cot t = \frac{\cos t}{\sin t}\). It is negative when the cosine and sine have opposite signs, which occurs in Quadrant II (where \(\cos t < 0\) and \(\sin t > 0\)) and Quadrant IV (where \(\cos t > 0\) and \(\sin t < 0\)).

Step 4: Determine the Intersection of Conditions

From Steps 2 and 3, we conclude that the condition \(\cos t < 0\) restricts \(t\) to Quadrants II and III, and \(\cot t < 0\) restricts \(t\) to Quadrants II and IV. Therefore, the common quadrant for both conditions is Quadrant II.

Key Concepts

Angles and Quadrants

Understanding angles and quadrants in a coordinate plane is fundamental to mastering trigonometry. A coordinate plane is divided into four quadrants, denoted by Roman numerals I, II, III, and IV:

• Quadrant I: Both x and y coordinates are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y coordinates are negative.
• Quadrant IV: x is positive, y is negative.

Each quadrant represents different sign combinations for the sine, cosine, and tangent functions of angles whose terminal sides lie within them. These sign variations are crucial for determining the properties and relationships of trigonometric functions. Recognizing which quadrant an angle terminates in can help identify the behavior and sign of trigonometric functions, such as cosine and cotangent, at that angle.

Cosine Function

The cosine function is one of the primary trigonometric functions, representing the x-coordinate of a point on a unit circle. In simpler terms, it helps describe an angle's relationship with the x-axis of the coordinate plane.

The cosine function, denoted as \(\cos t,\) measures the horizontal distance of a terminal point from the origin, with its range between -1 and 1. This means that:

• Positive Cosine Values: Occur in Quadrants I and IV, where the x-coordinate is positive.
• Negative Cosine Values: Occur in Quadrants II and III, where the x-coordinate is negative.

Understanding the behavior of the cosine function is essential for comprehending how angles relate to their respective positions on the coordinate plane. In our context, \(\cos t < 0\) indicates that the terminal point must lie in Quadrants II or III, as these are the regions where the x-coordinate is negative.

Cotangent Function

The cotangent function is another foundational concept in trigonometry and is denoted by\(\cot t.\) It is defined as the ratio of the cosine to the sine function, formulated as \(\cot t = \frac{\cos t}{\sin t}.\) Like cosine, cotangent also has sign constraints based on quadrant position.

The cotangent function can be expressed based on its sign in different quadrants:

• Positive Cotangent Values: Occur in Quadrants I and III, where both sine and cosine share the same sign.
• Negative Cotangent Values: Occur in Quadrants II and IV, where sine and cosine have opposite signs.

For a given angle \(\cot t < 0,\) the angle must terminate in either Quadrant II or IV. This implies that the cosine and sine of the angle must have opposite signs. In the problem context, combining this with the condition \(\cos t < 0\) leads to the consistent position of the terminal point in Quadrant II, where cosine is negative and sine is positive.