Problem 22 - Precalculus Essentials

Question

In Exercises \(17-22\), let \(\theta\) be an angle in standard position. Name the quadrant in which \(\theta\) lies $$\cot \theta>0, \quad \sec \theta<0$$

Step-by-Step Solution

Answer

The angle θ lies in the third quadrant.

Step 1: Determine the sign of cotangent

Cotangent is the reciprocal of tangent, so they share the same sign. Tangent is positive when sine and cosine have the same sign. In other words, we can say that cotangent is positive in the first and third quadrants.

Step 2: Determine the sign of secant

Secant is the reciprocal of cosine, so they share the same sign. Cosine is negative when the angle is in the second or third quadrant.

Step 3: Combining the information

From step 1, we have that θ might reside in first or third quadrant. From step 2, we have that θ might reside in second or third quadrant. Combining this information, the only quadrant that satisfies both conditions is the third quadrant.

Key Concepts

Cotangent

The cotangent function, expressed as \( \cot \theta \), is a fundamental trigonometric identity. It's defined as the reciprocal of the tangent function. This means:

• \( \cot \theta = \frac{1}{\tan \theta} \)

Since the tangent function is the ratio of sine to cosine, we also define cotangent in terms of sine and cosine:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

Cotangent shares the same sign as tangent, becoming positive when sine and cosine have the same sign.This occurrence happens in the first and third quadrants. If both sine and cosine are positive, as in the first quadrant, or both are negative, as in the third quadrant, then the cotangent is positive. This knowledge helps in identifying the quadrant in which an angle lies based on the cotangent value.

Secant

The secant function, noted as \( \sec \theta \), is another basic trigonometric identity, defined as the reciprocal of the cosine function. This relationship is expressed as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Because secant and cosine are reciprocals, they share the same sign.Cosine is positive in the first and fourth quadrants; thus, secant is also positive in these quadrants. Conversely, cosine turns negative in the second and third quadrants.Therefore, if \( \sec \theta < 0 \), the angle \( \theta \) must lie in either the second or third quadrant. Understanding secant's sign helps in understanding in which quadrant an angle may be based on secant's value.

Angle in Standard Position

An angle is said to be in standard position when its vertex is at the origin of the coordinate plane and its initial side lies along the positive x-axis. Here's a key point to remember:

• The amount of rotation from the initial side to the terminal side determines the size of the angle.

Angles can rotate in a counterclockwise direction, making them positive, or clockwise, making them negative. This standard positioning is critical for analyzing angles using trigonometric functions, as it provides a uniform way to visualize and interpret them. By using this uniform positioning, we can consistently apply trigonometric identities like cotangent and secant to determine the quadrant and nature of an angle.

Trigonometric Quadrants

The trigonometric quadrants divide the coordinate plane into four sections, each representing a range of angles. Each quadrant has distinct characteristics based on the signs of trigonometric functions:

• First Quadrant: Sine, cosine, and tangent are all positive.
• Second Quadrant: Sine is positive, while cosine and tangent are negative.
• Third Quadrant: Tangent is positive, whereas sine and cosine are negative.
• Fourth Quadrant: Cosine is positive, with sine and tangent negative.

These characteristics are crucial when analyzing angles in standard position and determining which trigonometric functions will exhibit positive or negative values based on the angle's quadrant.By knowing the behavior of trigonometric functions in these quadrants, one can accurately pinpoint angles based on given conditions, such as \( \cot \theta > 0 \) and \( \sec \theta < 0 \). In this example, \( \theta \) lies in the third quadrant, where both conditions are satisfied.