Problem 37 - Precalculus Mathematics for Calculus

Question

Find the quadrant in which \(\theta\) lies from the information given. $$\sec \theta>0 \text { and } \tan \theta<0$$

Step-by-Step Solution

Answer

\( \theta \) is in the 4th quadrant.

Step 1: Understanding the Trigonometric Values

To find the quadrant in which \( \theta \) lies, we need to analyze the signs of the given trigonometric functions. \( \sec \theta > 0 \) implies that \( \cos \theta > 0 \) because secant is the reciprocal of cosine. \( \tan \theta < 0 \) indicates that the sine and cosine have opposite signs, as tangent is the ratio of sine to cosine.

Step 2: Determining Cosine's Positive Quadrants

\( \cos \theta > 0 \) occurs in the 1st and 4th quadrants, as cosine is positive in these quadrants due to the x-coordinate being positive on the unit circle.

Step 3: Analyzing Tangent's Sign

\( \tan \theta < 0 \) means sine and cosine have opposite signs. The tangent function is negative when \( \sin \theta \) and \( \cos \theta \) have different signs, which happens in the 2nd and 4th quadrants.

Step 4: Finding The Common Quadrant

To satisfy both conditions \( \cos \theta > 0 \) and \( \tan \theta < 0 \), \( \theta \) must be in the 4th quadrant, since this is where cosine is positive and tangent is negative.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function. This means that \( \sec \theta = \frac{1}{\cos \theta} \). When we are told that \( \sec \theta > 0 \), it implies that \( \cos \theta > 0 \) as well. This is because dividing by a positive number (1 in this case) retains the sign of the original number.
In trigonometry, understanding where the secant is positive or negative helps determine the possible quadrants for the angle \( \theta \). Since secant is only defined when cosine is not zero, it also highlights the behavior of angles in relation to the x-axis of the unit circle. Therefore:

• \( \sec \theta > 0 \) in the first and fourth quadrants, since these are where \( \cos \theta \) is positive.
• Reciprocal identities like the secant are useful for solving more complex trigonometric problems when combined with other trigonometric information.

Tangent Function

The tangent function, \( \tan \theta \), is another key trigonometric function, defined as the ratio of sine to cosine. \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). When \( \tan \theta < 0 \), it implies that \( \sin \theta \) and \( \cos \theta \) have opposite signs. This is crucial for understanding their behavior in different quadrants.
This relationship between sine and cosine helps us determine where \( \tan \theta \) is negative:

• In the second quadrant, where sine is positive and cosine is negative, \( \tan \theta \) is negative.
• In the fourth quadrant, where sine is negative and cosine is positive, \( \tan \theta \) is negative as well.

Understanding where the tangent function is negative helps narrow down the possible locations for \( \theta \) on the unit circle. It provides a strategy for solving trigonometric problems involving angles and their corresponding function values.

Sine and Cosine Signs

The signs of sine (\( \sin \theta \)) and cosine (\( \cos \theta \)) are critical in determining the quadrant in which an angle \( \theta \) lies. These signs vary depending on which quadrant the angle is located in on the unit circle.

• First Quadrant: Both \( \sin \theta \) and \( \cos \theta \) are positive.
• Second Quadrant: \( \sin \theta \) is positive and \( \cos \theta \) is negative.
• Third Quadrant: Both \( \sin \theta \) and \( \cos \theta \) are negative.
• Fourth Quadrant: \( \sin \theta \) is negative and \( \cos \theta \) is positive.

These sign changes are due to the circular nature of trigonometric functions and relate to the x- and y-coordinates of angles on the unit circle. By knowing which trig function is positive or negative, one can deduce the placement of \( \theta \) within the specific quadrants, providing a deeper understanding necessary for solving related trigonometric equations or inequalities.