Problem 22 - Algebra and Trigonometry

Question

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 \(\theta\) lies in the third quadrant.

Step 1: Define the Problem

Given that \(\cot \theta>0\) and \(\sec \theta<0\), the objective is to find the quadrant in which the angle \(\theta\) falls.

Step 2: Analyze the Cotangent

We know that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). The cotangent is positive when both cosine and sine are either both positive or both negative. If \(\cot \theta>0\), then that means cosine and sine must either both be positive or both be negative. This occurs in the first quadrant (where all trigonometric values are positive) and the third quadrant (where both cosine and sine are negative).

Step 3: Analyze the Secant

Next, it is given that \(\sec \theta<0\). Recall that \(\sec \theta = \frac{1}{\cos \theta}\). Therefore, if \(\sec \theta<0\), that means that \(\cos \theta\) must be negative. Cosine is negative in the second and third quadrants.

Step 4: Find the Overlapping Quadrant

Lastly, it is necessary to find a quadrant that satisfies both conditions. From step 2, either the first or third quadrant could satisfy \(\cot \theta>0\), and from step 3, either the second or third quadrant could satisfy \(\sec \theta<0\). Therefore, the only quadrant that fits both conditions is the third quadrant.

Key Concepts

Cotangent

The cotangent function, denoted as \( \cot \theta \), represents the ratio of the cosine to the sine of an angle \( \theta \).
This is expressed mathematically as:

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

This essential trigonometric function behaves uniquely in the four standard quadrants of the Cartesian plane.

- In the first quadrant, both \( \cos \theta \) and \( \sin \theta \) are positive, making \( \cot \theta \) positive.- In the second quadrant, \( \cos \theta \) is negative, and \( \sin \theta \) is positive, resulting in a negative cotangent.- In the third quadrant, both \( \cos \theta \) and \( \sin \theta \) are negative, and \( \cot \theta \) is thus positive.- Lastly, in the fourth quadrant, \( \cos \theta \) is positive while \( \sin \theta \) is negative, leading to a negative cotangent.

Understanding when \( \cot \theta \) is positive or negative is crucial since it informs us about the possible quadrants \( \theta \) could lie in during trigonometric problem-solving.

Secant

The secant function, represented as \( \sec \theta \), is a reciprocal trigonometric function.
It is the inverse of the cosine function, calculated as:

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

This relationship means that the secant function takes on the sign of the cosine function.

- In the first quadrant, \( \cos \theta \) is positive, making \( \sec \theta \) positive.- In the second quadrant, \( \cos \theta \) turns negative, causing \( \sec \theta \) to also be negative.- The trend continues: in the third quadrant, \( \cos \theta \) is negative, so \( \sec \theta \) is negative as well.- Finally, in the fourth quadrant, \( \cos \theta \) returns to a positive value, leading to a positive \( \sec \theta \).

Hence, for situations where \( \sec \theta < 0 \), the angle \( \theta \) could only lie in the second or third quadrants.
Recognizing where \( \sec \theta \) is negative is important for determining the specific quadrant for \( \theta \).

Quadrants

The Cartesian plane consists of four quadrants.
When discussing the angle \( \theta \), it's important to understand how trigonometric functions behave differently in each.

• The first quadrant (I) has all positive trigonometric functions: \( \sin \theta > 0 \), \( \cos \theta > 0 \), and \( \tan \theta > 0 \).
• In the second quadrant (II), \( \sin \theta > 0 \) while \( \cos \theta < 0 \) and \( \tan \theta < 0 \).
• The third quadrant (III) is characterized by all trigonometric ratios being negative except \( \tan \theta > 0 \), thus \( \sin \theta < 0 \) and \( \cos \theta < 0 \).
• In the fourth quadrant (IV), \( \cos \theta > 0 \) while \( \sin \theta < 0 \) and \( \tan \theta < 0 \).

Finding the correct quadrant involves identifying the signs of trigonometric functions for \( \theta \).
This particular exercise demonstrates how the conditions \( \cot \theta > 0 \) and \( \sec \theta < 0 \) limit \( \theta \) to the third quadrant.
Grasping these relationships helps determine where an angle can fall based on trigonometric constraints.