Problem 95 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The angle \( \theta \) is in the third or fourth quadrant.

Step 1: Understanding the Condition \\ \sin \theta < 0

The sine function is negative in the third and fourth quadrants. This is because in these quadrants, the y-coordinate (the opposite side in a unit circle) is negative, making the sine, which represents the y-coordinate, also negative.

Step 2: Understanding the Condition \\ \csc \theta < 0

The cosecant function is the reciprocal of the sine function: \( \csc \theta = \frac{1}{\sin \theta} \). This means that \( \csc \theta < 0 \) wherever \( \sin \theta < 0 \), which are the third and fourth quadrants, confirming the consistency in these locations for both conditions.

Step 3: Determine the Quadrants Based on Both Conditions

Since both \( \sin \theta < 0 \) and \( \csc \theta < 0 \) occur in the third and fourth quadrants, \( \theta \) must lie in one or both of these quadrants.

Key Concepts

Quadrants of an Angle

Angles in the Cartesian coordinate system are divided into four quadrants. Each quadrant exhibits unique characteristics due to the coordinate signs. Here's a simple way to understand the quadrants:

• First Quadrant (QI): Both x and y coordinates are positive. This makes sine and cosine values positive as well.
• Second Quadrant (QII): The x coordinate is negative, while the y coordinate remains positive. Sine values stay positive, but cosine values turn negative.
• Third Quadrant (QIII): Both x and y coordinates are negative. Therefore, both sine and cosine values become negative.
• Fourth Quadrant (QIV): The x coordinate is positive, but the y coordinate is negative. This results in positive cosine values and negative sine values.

Whenever dealing with trigonometric functions, understanding which quadrant the angle lies in helps to determine the sign of trigonometric functions like sine or cosine.

Sine Function

The sine function is a foundational concept in trigonometry. It relates to the y-coordinate on the unit circle. Here's a bit more to make sense of it:

• Definition: Sine of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Unit Circle: On the unit circle, \( \sin \theta \) corresponds to the y-coordinate of the point where the terminal side of \( \theta \) intersects the unit circle.
• Sign Changes: Sine is positive in both the first and second quadrants, where the y-coordinate is positive. However, it becomes negative in the third and fourth quadrants, as the y-coordinate is negative there.

In our given problem, because \( \sin \theta < 0 \), \( \theta \) must lie in the third or fourth quadrant where the sine values turn negative.

Cosecant Function

The cosecant function (\( csc \theta \)) is the reciprocal of the sine function. It is crucial to understand its relationship with sine to tackle problems related to it:

• Definition: Cosecant is defined as the ratio of the hypotenuse to the opposite side. In formula form, \( \csc \theta = \frac{1}{\sin \theta} \).
• Behavior: Since cosecant is directly tied to the sine function, it follows that \( \csc \theta \) shares the same sign as \( \sin \theta \). Therefore, wherever sine is negative, cosecant will also be negative.
• Quadrant Implications: For the condition \( \csc \theta < 0 \) to hold true, \( \theta \) must be in quadrants where sine is negative, which are the third and fourth quadrants.

In this exercise, \( \csc \theta < 0 \) is true in the same quadrants as \( \sin \theta < 0 \), reaffirming that \( \theta \) must lie in the third or fourth quadrant. Understanding this interconnectedness helps solve trigonometric problems more effectively.