Problem 35 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Properties of Trigonometric Functions

The sine function, \(\sin \theta\), is positive in the first and second quadrants, and negative in the third and fourth quadrants. The cosine function, \(\cos \theta\), is positive in the first and fourth quadrants, and negative in the second and third quadrants.

Step 2: Analyze the Given Condition

We are given that \(\sin \theta < 0\). This means \(\theta\) is either in the third or fourth quadrant. Additionally, \(\cos \theta < 0\), which indicates \(\theta\) is either in the second or third quadrant.

Step 3: Identify the Overlapping Quadrant

The overlap from both conditions where \(\sin \theta < 0\) and \(\cos \theta < 0\) is the third quadrant. This is the only quadrant where both sine and cosine values are negative.

Key Concepts

Sine Function

The sine function, often represented as \( \sin \theta \), is one of the fundamental trigonometric functions. It relates the angle \( \theta \) in a right triangle to the ratio of the length of the opposite side over the hypotenuse. This function is widely used to determine the vertical position relative to the unit circle.
Understanding the sine function's behavior in different quadrants is crucial for solving trigonometric problems. In the first quadrant, both sine and cosine are positive. Moving to the second quadrant, sine remains positive, but cosine becomes negative. In the third and fourth quadrants, the sine function turns negative.

• First Quadrant: \( \sin \theta > 0 \)
• Second Quadrant: \( \sin \theta > 0 \)
• Third Quadrant: \( \sin \theta < 0 \)
• Fourth Quadrant: \( \sin \theta < 0 \)

The sine function's sign change across the quadrants helps to determine which quadrant an angle \( \theta \) belongs to when combined with cosine.

Cosine Function

The cosine function, denoted as \( \cos \theta \), is also a primary trigonometric function. It represents the ratio of the length of the adjacent side over the hypotenuse in a right triangle. The cosine function provides insight into the horizontal position on the unit circle.
The sign of the cosine function varies depending on the quadrant you are considering. In the first quadrant, cosine, like sine, is positive. As you move to the second quadrant, cosine turns negative, remaining negative in the third quadrant as well. In the fourth quadrant, cosine becomes positive again.

• First Quadrant: \( \cos \theta > 0 \)
• Second Quadrant: \( \cos \theta < 0 \)
• Third Quadrant: \( \cos \theta < 0 \)
• Fourth Quadrant: \( \cos \theta > 0 \)

Recognizing how the cosine function changes sign across quadrants is useful in pinpointing the exact location of an angle in the coordinate plane.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants, each offering different characteristics for trigonometric functions. Understanding these quadrants makes it easier to predict the behavior of sine and cosine values for various angles \( \theta \).
The quadrants are labeled in a counterclockwise direction:

• First Quadrant: Here, both \( \sin \theta \) and \( \cos \theta \) are positive.
• Second Quadrant: \( \sin \theta \) is positive while \( \cos \theta \) turns negative.
• Third Quadrant: Both \( \sin \theta \) and \( \cos \theta \) are negative, aligning with the solution to the original exercise.
• Fourth Quadrant: \( \sin \theta \) is negative, but \( \cos \theta \) is positive again.

These quadrants help in determining the signs of the trigonometric functions and solving equations such as finding the quadrant where both \( \sin \theta \) and \( \cos \theta \) are negative, leading us directly to the third quadrant.