Problem 16 - Precalculus

Question

Indicate the quadrant in which the terminal side of \(\theta\) must lie in order for the information to be true. \(\cos \theta\) is negative and \(\sin \theta\) is positive.

Step-by-Step Solution

Answer

The terminal side of \( \theta \) is in Quadrant II.

Step 1: Understanding the problem

We need to determine in which quadrant the angle \( \theta \) lies based on the conditions: \( \cos \theta \) is negative and \( \sin \theta \) is positive.

Step 2: Recall quadrant properties

Recall that: - In Quadrant I, both sine and cosine are positive. - In Quadrant II, sine is positive and cosine is negative. - In Quadrant III, both sine and cosine are negative. - In Quadrant IV, sine is negative and cosine is positive.

Step 3: Identify the correct quadrant

Based on the given conditions:- \( \cos \theta < 0 \) and \( \sin \theta > 0 \),only Quadrant II fits both these conditions because cosine is negative and sine is positive there.

Key Concepts

Understanding Quadrants

When working with trigonometric functions, understanding the quadrants on the Cartesian plane is crucial. The coordinate plane is divided into four quadrants:

• Quadrant I: Located in the upper right part of the plane. Here, both the sine and cosine values are positive.
• Quadrant II: Situated in the upper left part. In this quadrant, sine is positive, but cosine is negative.
• Quadrant III: Found in the lower left section. Both sine and cosine are negative here.
• Quadrant IV: Located in the lower right. Sine is negative and cosine is positive.

Starting from the positive x-axis, the quadrants move counterclockwise from I to IV. Each quadrant has distinct characteristics that align with sine and cosine values. Knowing these properties is essential when solving problems related to angles and trigonometric functions.
Identifying the correct quadrant based on conditions is often the first step in solving trigonomic problems.

Exploring Cosine

Cosine, often noted as \( \cos \), is a fundamental trigonometric function. It relates the angle of a right triangle to the adjacent side and the hypotenuse. In the coordinate system, cosine represents the x-coordinate of a point on the unit circle.
Here are some core points about cosine:

• The value of cosine ranges between -1 and 1.
• Cosine is positive in Quadrants I and IV.
• Cosine becomes negative in Quadrants II and III.

Understanding the behavior of cosine across different quadrants is key when solving trigonometric equations. For example, if the problem states that \( \cos \theta \) is negative, it implies that the angle is either in Quadrant II or III.
This identification helps narrow down possibilities when solving trigonomic problems with multiple potential solutions.

Unpacking Sine

Sine, represented by \( \sin \), is another primary trigonometric function. It defines the relationship between the angle of a right triangle and the opposite side to the hypotenuse. Within the circular coordinate system, sine measures the y-coordinate of a point on the unit circle.
Some important sine attributes include:

• The range of sine is from -1 to 1.
• Sine is positive in Quadrants I and II.
• In Quadrants III and IV, sine is negative.

When solving trigonometry problems, the sign of sine indicates possible quadrant locations. For instance, the statement \( \sin \theta > 0 \) points to either Quadrant I or II.
Such understanding is helpful to evaluate conditions given in trigonomic contexts, enabling identification of quadrant placement for angles efficiently.