Problem 22 - Precalculus: A Concise Course

Question

State the quadrant in which \(\theta\) lies. $$ \sec \theta>0 \text { and } \cot \theta<0 $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the signs of trigonometric functions in each quadrant

Remember that in the first quadrant (0 to 90 degrees) all trigonometric functions are positive, in the second quadrant (90 to 180 degrees) only \(\sin\) and its reciprocal function are positive, in the third quadrant (180 to 270 degrees) only \(\tan\) and its reciprocal function are positive and in the fourth quadrant (270 to 360 degrees) only \(\cos\) and its reciprocal function are positive.

Step 2: Apply the conditions in the problem

The problem states that \(\sec \(\theta\) > 0\) and \(\cot \(\theta\) < 0\). \(\sec\) is the reciprocal of \(\cos\) and is positive in the first and fourth quadrants. \(\cot\) is the reciprocal of \(\tan\) and is negative in the second and fourth quadrants.

Step 3: Conclude the quadrant

By satisfying both conditions, the only quadrant that fulfills both conditions is the fourth quadrant. Therefore, the angle \(\theta\) lies in the fourth quadrant.

Key Concepts

Secant Function

The secant function, often denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, meaning \( \sec \theta = \frac{1}{\cos \theta} \). Understanding its properties can help in solving various trigonometric problems. Here are some key points about the secant function:

• Reciprocal Nature: Since it is the reciprocal of cosine, where cosine equals zero, the secant function is undefined. These points correspond to angles of \(90^\circ\) and \(270^\circ\) or \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \) in radians.
• Positivity in Quadrants: The secant function is positive in quadrants where the cosine function is positive—specifically, the first and fourth quadrants. In these regions, the angle's cosine is greater than zero.
• Application: The secant function is particularly useful in solving problems involving right triangles and in calculus, especially in the context of integrals and derivatives.

Knowing when \( \sec \theta \) is positive helps determine angle quadrants, as seen in the provided exercise.

Cotangent Function

The cotangent function, represented as \( \cot \theta \), is another important trigonometric function. It is the reciprocal of the tangent function, or \( \cot \theta = \frac{1}{\tan \theta} \). This function plays a crucial role in understanding the relationships between the angles and their trigonometric values. Key characteristics include:

• Reciprocal Nature: Because it is the reciprocal of tangent, \( \cot \theta \) is undefined when \( \tan \theta = 0 \), which occurs at angles of \(0^\circ\), \(180^\circ\), and \(360^\circ\) (\(0\), \(\pi\), and \(2\pi\) radians).
• Significance of Sign: The cotangent is positive where the sine and cosine functions have the same sign (first and third quadrants). It becomes negative where they have opposite signs (second and fourth quadrants).
• Practical Utility: The cotangent is often useful in the analysis of angular measurements, particularly in spherical trigonometry and in solving certain calculus problems.

In the context of the exercise, knowing \( \cot \theta \) is negative helps us deduce the quadrant in which \( \theta \) lies.

Angle Quadrants

Angles in trigonometry are divided into four quadrants, each 90 degrees (or \(\pi/2\) radians) wide, as part of the Cartesian plane. Understanding which quadrant an angle lies in can reveal crucial information about the sign and values of its trigonometric functions.

• First Quadrant (0 to 90 degrees or 0 to \(\pi/2\)): All trigonometric functions (sine, cosine, tangent, secant, cosecant, cotangent) are positive.
• Second Quadrant (90 to 180 degrees or \(\pi/2\) to \(\pi\)): Sine and cosecant are positive while the others are negative.
• Third Quadrant (180 to 270 degrees or \(\pi\) to \(3\pi/2\)): Tangent and cotangent are positive, while sine, cosine, secant, and cosecant are negative.
• Fourth Quadrant (270 to 360 degrees or \(3\pi/2\) to \(2\pi\)): Cosine and secant are positive, whereas the other functions are negative.

In the exercise, the angle \( \theta \) was established to lie in the fourth quadrant based on the conditions \( \sec \theta > 0 \) and \( \cot \theta < 0 \), aligning with the quadrant characteristics.