Problem 38 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

Theta is in the second quadrant.

Step 1: Recall Definition of Cosecant

The cosecant function, \(\csc \theta\), is the reciprocal of the sine function. Hence, \(\csc \theta > 0\) implies that \(\sin \theta > 0\).

Step 2: Identify Sine Positive Quadrants

The sine function is positive in the first and second quadrants. This tells us that \(\theta\) must be in either the first or second quadrant.

Step 3: Recall Definition of Cosine

The cosine function, \(\cos \theta\), is positive in the first and fourth quadrants but negative in the second and third quadrants. Since \(\cos \theta < 0\), this implies \(\theta\) must be in the second or third quadrant.

Step 4: Determine the Quadrant

Combine the findings from Steps 2 and 3: \(\sin \theta > 0\) suggests the first or second quadrant, and \(\cos \theta < 0\) suggests the second or third quadrant. The only quadrant satisfying both conditions is the second quadrant.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function. This means: \[ \csc \theta = \frac{1}{\sin \theta} \] This definition implies some important characteristics:

• If \( \csc \theta > 0 \), then \( \sin \theta > 0 \).
• Conversely, if \( \csc \theta < 0 \), then \( \sin \theta < 0 \).

Since cosecant is undefined when sine is zero, it highlights points where sine crosses an axis. These concepts are crucial in understanding the behavior of trigonometric functions based in different quadrants. In particular situations, such as finding which quadrant a specific angle \( \theta \) lies in, knowing these reciprocals helps determine which other trigonometric functions might be positive or negative. This can tell us significant details about the angle and its position on the unit circle.

Sine Function

The sine function, denoted as \( \sin \theta \), is fundamental in trigonometry, describing the ratio of the side of a right triangle opposite the angle \( \theta \) to the hypotenuse. In a unit circle, \( \sin \theta \) represents the y-coordinate of a point formed by an angle \( \theta \) from the positive x-axis. The sign of \( \sin \theta \) helps us identify which quadrant \( \theta \) is in:

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

So, if \( \sin \theta > 0 \), as with the given exercise conditions, \( \theta \) must be in the first or second quadrants. This behavior is consistent with the general properties of sine across the four quadrants in the unit circle. Understanding these properties is essential to solving problems in trigonometry related to angles and their positions.

Cosine Function

The cosine function, represented as \( \cos \theta \), provides the ratio of the adjacent side over the hypotenuse in a right triangle. Within a unit circle, \( \cos \theta \) signifies the x-coordinate of the point on the circle, corresponding to the angle \( \theta \). About quadrants, the cosine function shows these characteristics:

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

Given \( \cos \theta < 0 \) in our exercise, \( \theta \) must fall in the second or third quadrant. The negative value of \( \cos \theta \) is crucial as it helps differentiate between quadrants when combined with sine's positive or negative values. Such knowledge allows for determining more defined locations of angles based on cosine values, assisting in solving broader trigonometric problems.