Problem 19 - Precalculus

Question

Indicate the quadrant in which the terminal side of \(\theta\) must lie in order for the information to be true. \(\sec \theta\) and \(\csc \theta\) are both positive.

Step-by-Step Solution

Answer

The angle \( \theta \) lies in Quadrant I.

Step 1: Understand the Trigonometric Functions

The secant function \( \sec \theta \) is the reciprocal of the cosine function, \( \cos \theta \). When \( \sec \theta \) is positive, this means \( \cos \theta \) is also positive. Similarly, the cosecant function \( \csc \theta \) is the reciprocal of the sine function, \( \sin \theta \). For \( \csc \theta \) to be positive, \( \sin \theta \) must also be positive.

Step 2: Determine Where Cosine is Positive

Cosine is positive in Quadrants I and IV. This is because in these quadrants, the x-coordinate on the unit circle, which corresponds to the cosine value, is positive.

Step 3: Determine Where Sine is Positive

Sine is positive in Quadrants I and II. This corresponds to the y-coordinate on the unit circle, which is positive in these quadrants.

Step 4: Find the Quadrant Where Both are Positive

Looking at the information from the previous steps, both \( \cos \theta \) and \( \sin \theta \) are positive only in Quadrant I. This is the only quadrant where both trigonometric functions are positive simultaneously.

Key Concepts

Quadrants

In trigonometry, the concept of quadrants is crucial for understanding the sign of trigonometric functions like \( \cos \ heta \) and \( \sin \ heta \). The coordinate plane is divided into four quadrants, numbered counterclockwise:

• Quadrant I: Both x and y coordinates are positive.
• Quadrant II: x is negative, and y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, and y is negative.

Each quadrant has specific properties that determine the sign of the cosine and sine functions. Knowing these properties helps determine where a given angle’s terminal side will fall based on the signs of its trigonometric functions.

Cosine

The cosine function, represented as \( \cos \ heta \), expresses the x-coordinate on the unit circle corresponding to an angle \( \theta \). Cosine is directly linked to how far right or left a point is on a unit circle.

• In Quadrants I and IV, the x-coordinate is positive, so \( \cos \ heta \) is positive.
• In Quadrants II and III, the x-coordinate is negative, making \( \cos \ heta \) negative.

Understanding the cosine function's behavior in different quadrants is essential for analyzing angles and understanding trigonometric relationships.

Sine

The sine function, expressed as \( \sin \ heta \), represents the y-coordinate of an angle \( \theta \) on the unit circle. It dictates how high or low a point is positioned vertically.

• In Quadrants I and II, the y-coordinate is positive, meaning \( \sin \ heta \) is positive.
• In Quadrants III and IV, the y-coordinate is negative, so \( \sin \ heta \) is negative.

This means that for both \( \sin \ heta \) and \( \cos \ heta \) to be positive, the angle must be in Quadrant I.

Unit Circle

The unit circle is a fundamental concept that aids in understanding trigonometric functions. It is a circle with radius 1 centered at the origin of a coordinate plane.
Angles are measured from the positive x-axis counterclockwise. Each point on the unit circle has coordinates \( (\cos \ heta, \sin \ heta) \), making it instrumental in visualizing how the sine and cosine functions change with different angles. The unit circle helps with:

• Determining the sign of trigonometric functions in different quadrants.
• Visualizing the position of angles and their terminal sides.
• Preferring conceptual understanding over memorization of values.

Using the unit circle, determining which quadrant an angle lies in becomes intuitive, helping solve problems involving trigonometric functions effectively.