Problem 15 - 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 positive and \(\sin \theta\) is negative.

Step-by-Step Solution

Answer

The terminal side of \(\theta\) lies in Quadrant IV.

Step 1: Understanding Angles in Quadrants

The coordinate plane is divided into four quadrants. Quadrant I is where both sine and cosine are positive. Quadrant II has cosine negative, but sine positive. Quadrant III has both sine and cosine negative. Quadrant IV has cosine positive and sine negative.

Step 2: Apply Quadrant Characteristics

According to the problem, \(\cos \theta\) is positive and \(\sin \theta\) is negative. Based on the quadrant characteristics explained in Step 1, we see that only Quadrant IV fulfills these conditions.

Step 3: Conclusion

From the analysis, the terminal side of \(\theta\) must lie in Quadrant IV to have \(\cos \theta\) positive and \(\sin \theta\) negative.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface defined by an X-axis and a Y-axis. It is used to plot points and understand positions in mathematics. The axes divide the plane into four separate areas known as quadrants. Each point on the plane can be represented by an ordered pair \(x, y\). Here, the x-value indicates the point's position relative to the Y-axis, and the y-value shows its position relative to the X-axis. In terms of equations and graphs, the coordinate plane helps visualize complex functions and relationships. It provides a visual method to understand concepts in algebra, geometry, and trigonometry. Each position, derived from the coordinates, reflects a unique part of the mathematical graph and is crucial in understanding angles and trigonometric functions like sine and cosine. Understanding the layout of this plane is an essential foundation for solving problems involving trigonometric functions.

Quadrants

The coordinate plane is divided into four distinct quadrants, providing a framework to classify the location of angles and points based on their coordinates:

• Quadrant I: Both the x and y coordinates are positive. This means both sine and cosine values are positive as well.
• Quadrant II: The x coordinate is negative, while the y coordinate is positive. Therefore, sine values are positive, but cosine values are negative.
• Quadrant III: Both the x and y coordinates are negative. Here, both sine and cosine values are negative.
• Quadrant IV: The x coordinate is positive, and the y coordinate is negative. This results in positive cosine values, but negative sine values.

Each quadrant reflects different characteristics for trigonometric functions, determined by the signs of the coordinates. Recognizing these characteristics informs how angles will behave within each quadrant and is vital for identifying where specific trigonometric values occur.

Cosine and Sine Values in Quadrants

The values of the cosine and sine functions are closely linked to their positions within the quadrants of the coordinate plane. Cosine represents the horizontal coordinate (x-value) of an angle’s terminal side when plotted from the origin. Conversely, sine denotes the vertical coordinate (y-value). Hence, the quadrant decides the sign.

• In Quadrant I, both cosine and sine are positive, as both coordinates are positive.
• In Quadrant II, cosine is negative, sine is positive due to the negative x coordinate and positive y coordinate.
• In Quadrant III, both functions are negative, with both coordinates being negative.
• In Quadrant IV, cosine is positive, sine is negative since the x coordinate is positive and the y coordinate is negative.

Understanding these relationships is essential for solving problems involving trigonometric conditions, such as identifying the quadrant where certain conditions like positive cosine and negative sine fit, which in this case, is Quadrant IV. These insights allow for a more straightforward interpretation of sine and cosine values based on their locations in quadrants.