Problem 93 - A Graphical Approach to Precalculus with Limits

Question

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\sec \theta<0, \csc \theta<0$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Conditions for Secant

The secant function, \( \sec \theta = \frac{1}{\cos \theta} \), is negative when \( \cos \theta \) is negative. The cosine function is negative in the second and third quadrants.

Step 2: Understand the Conditions for Cosecant

The cosecant function, \( \csc \theta = \frac{1}{\sin \theta} \), is negative when \( \sin \theta \) is negative. The sine function is negative in the third and fourth quadrants.

Step 3: Determine the Overlapping Quadrant

Both conditions need to be satisfied simultaneously: \( \cos \theta \) is negative in the second and third quadrants, while \( \sin \theta \) is negative in the third and fourth quadrants. The only quadrant where both \( \cos \theta \) and \( \sin \theta \) are negative is the third quadrant.

Key Concepts

Quadrants

In trigonometry, a quadrant refers to one of the four sections of the coordinate plane. These sections are divided by the x-axis and y-axis. Each quadrant represents a specific range of angle measures:

• First Quadrant (QI): Angles between 0° and 90°. Here, both sine and cosine are positive.
• Second Quadrant (QII): Angles between 90° and 180°. In this quadrant, sine is positive, but cosine is negative.
• Third Quadrant (QIII): Angles between 180° and 270°. Both sine and cosine values are negative here.
• Fourth Quadrant (QIV): Angles between 270° and 360°. Here, cosine is positive, but sine is negative.

Understanding the quadrants is essential for identifying where certain angle conditions are satisfied. They help determine the signs for trigonometric functions in each quadrant, aiding in exercises like identifying specific angle solutions.

Secant Function

The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function. It is defined as:\[ \sec \theta = \frac{1}{\cos \theta} \]Secant is less commonly used than sine or cosine but plays a crucial role in certain trigonometric applications. It is undefined whenever the cosine of the angle is zero because division by zero is undefined:

• Secant is positive where \text{cos\( \theta \)} is positive (in the first and fourth quadrants).
• Secant is negative where \text{cos\( \theta \)} is negative (in the second and third quadrants).

When dealing with angles, knowing where the secant function is negative helps pinpoint which quadrants are valid solutions for specific conditions.

Cosecant Function

The cosecant function, represented as \( \csc \theta \), is the reciprocal of the sine function and is defined by:\[ \csc \theta = \frac{1}{\sin \theta} \]Similar to the secant function, cosecant is less commonly employed but is vital in particular trigonometric analyses. It is undefined when the sine of an angle is zero. The behavior of cosecant is:

• Positive where \text{sin\( \theta \)} is positive (in the first and second quadrants).
• Negative where \text{sin\( \theta \)} is negative (in the third and fourth quadrants).

Recognizing which quadrants the cosecant function is negative is invaluable for addressing exercises involving specific angle properties.

Cosine

The cosine function, denoted by \( \cos \theta \), measures the horizontal component of an angle in a right triangle or the x-coordinate on the unit circle.The cosine value dictates the sign and vertical positioning of trigonometric functions in various quadrants:

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

The cosine function is crucial for analyzing the secant function, as \[ \sec \theta = \frac{1}{\cos \theta} \]Understanding these relationships is essential for solving problems that depend on the sign of the cosine.

Sine

The sine function is represented by \( \sin \theta \) and measures the vertical component of an angle in a right triangle or the y-coordinate of the angle when plotted on the unit circle.

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

This function is fundamental for the cosecant function, as the negative or positive sign indicates where the \[ \csc \theta = \frac{1}{\sin \theta} \]is valid.By understanding the sine function's behavior in different quadrants, you can resolve complex trigonometric conditions efficiently.