Problem 54 - Algebra and Trigonometry

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 the second quadrant.

Step 1: Analyze the Sign of Cosine

The cosine function is negative in the second and third quadrants. This gives us information about the possible quadrants where the terminal point could lie.

Step 2: Analyze the Sign of Cotangent

The cotangent function is negative when the sine and cosine have opposite signs. Therefore, cotangent is negative in the second quadrant (where sine is positive and cosine is negative) and in the fourth quadrant (where sine is negative and cosine is positive).

Step 3: Determine the Common Quadrant

From Steps 1 and 2, we know that cosine is negative in the second and third quadrants, and cotangent is negative in the second and fourth quadrants. The common quadrant for both conditions is the second quadrant.

Key Concepts

Quadrants

In trigonometry, the coordinate plane is divided into four sections called quadrants. Each quadrant corresponds to different ranges of angles and determines the signs of trigonometric functions. These quadrants are labeled counterclockwise, starting from the positive x-axis:

• First Quadrant: Angles from 0° to 90°
• Second Quadrant: Angles from 90° to 180°
• Third Quadrant: Angles from 180° to 270°
• Fourth Quadrant: Angles from 270° to 360°

In each quadrant, the sign of the trigonometric functions varies:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, whereas cosine and tangent are negative.
• Quadrant III: Tangent is positive, while sine and cosine are negative.
• Quadrant IV: Cosine is positive, but sine and tangent are negative.

Understanding which quadrant an angle falls into helps us determine the signs of trigonometric functions and solve problems related to angles and the unit circle.

Cosine

Cosine is a fundamental trigonometric function that indicates the ratio of the adjacent side to the hypotenuse in a right triangle. When considering the unit circle, cosine of an angle describes the x-coordinate of the corresponding point on the circle.

In terms of quadrants:

• Cosine is positive in the first and fourth quadrants.
• Cosine is negative in the second and third quadrants.

This characteristic of cosine is essential when solving problems related to angle determination on the coordinate plane. For instance, if you're told that cosine of an angle is negative, you can deduce that the terminal side of the angle lies in either the second or third quadrant. Recognizing these positive or negative affiliations helps comprehensively approach and solve trigonometric queries.

Cotangent

Cotangent is another trigonometric function known as the reciprocal of tangent. The formula for cotangent is: \[\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\]This tells us that cotangent is calculated by dividing the cosine of an angle by its sine.

Furthermore, the sign of cotangent depends on the signs of sine and cosine:

• Cotangent is positive when both sine and cosine are either positive or negative.
• Cotangent is negative when sine and cosine have opposite signs.

Hence, when \(\cot \theta\) is negative, and knowing that cosine is negative, you find that sine must be positive for cotangent to be negative. This situation occurs in the second quadrant. Understanding these relationships aids in determining the position of angles and in solving trigonometric equations effectively.