Problem 64 - Algebra 2

Question

An angle drawn in standard position has a terminal side that passes through the point \((\sqrt{2},-\sqrt{2}) .\) What is one possible measure of the angle? $$ \begin{array}{lllll}{\text { F. } 45^{\circ}} & {\text { G. } 225^{\circ}} & {\text { H. } 315^{\circ}} & {\text { J. } 330^{\circ}}\end{array} $$

Step-by-Step Solution

Answer

The possible measure of the angle in standard position whose terminal side that passes through the point (\(\sqrt{2}, -\sqrt{2}\)) is \(315^{\circ}\).

Step 1: Understand the problem

The problem is asking to find the measure of an angle whose terminal side passes through the point (\(\sqrt{2}, -\sqrt{2}\)). The angle is drawn in standard position which means that the initial side lies along the positive x-axis.

Step 2: Locate the point on the Cartesian coordinate system

Plot the point (\(\sqrt{2}, -\sqrt{2}\)) on the coordinate system. You will notice that the point is in the fourth quadrant.

Step 3: Determine the angle

Since the point is in the fourth quadrant, the terminal side of our angle will end up in the fourth quadrant as well. The measure of the angle in the fourth quadrant is \(360^{\circ} - \theta\), \(\theta\) is the acute angle made by the terminal side with the x-axis. Since the y-coordinate is negative and the x-coordinate is positive, both have the same absolute values, the angle formed (\(\theta\)) is \(45^{\circ}\). Hence, the measure of our angle (in standard position) is \(360^{\circ} - 45^{\circ} = 315^{\circ}\).

Key Concepts

Cartesian Coordinate System

The Cartesian Coordinate System is a two-dimensional plane defined by the x-axis and the y-axis.
The x-axis runs horizontally, and the y-axis runs vertically, intersecting at the origin (0,0). This system helps us plot points using pairs of coordinates \(x, y\), where each value indicates the point's distance from the respective axis.

To understand this further, let's visualize a graph:

• The horizontal line represents the x-axis.
• The vertical line represents the y-axis.
• The point where these lines intersect is the origin, marked as (0,0).

This system divides the plane into four quadrants, numbered counterclockwise starting from the upper right. Each quadrant helps us identify where points are located based on the signs of their x and y values.

Fourth Quadrant

The Fourth Quadrant is one of the four divisions of the Cartesian Coordinate System.
It is located in the bottom-right corner of the graph.
In the fourth quadrant:

• The x-values are positive because they are to the right of the origin.
• The y-values are negative because they are below the origin.

This quadrant is crucial when working with standard position angles, especially when identifying where the terminal side of an angle might lie. For instance, the terminal side might pass through points like \(\sqrt{2}, -\sqrt{2}\) , indicating a fourth quadrant angle.

Angle Measure

When we talk about Angle Measure in the context of the Cartesian Coordinate System, we refer to the degree of rotation from the positive x-axis to the terminal side of the angle.
Angles in standard position start with their initial side lying along the positive x-axis.

Key points:

• Angles measured counterclockwise from the positive x-axis are positive.
• Angles measured clockwise are negative.
• A full rotation is \(360^\circ\).

Within each quadrant, angles can vary. For instance, in the fourth quadrant, angles are often calculated by subtracting the acute angle from \(360^\circ\) to find the total angle measure.

Terminal Side of an Angle

The Terminal Side of an Angle is the side that moves or rotates from its initial position along the positive x-axis in standard position.
After this rotation, the terminal side ends up in one of the four quadrants, indicating the angle's position and measure.

For example, in a standard position where you need to find the angle passing through \(\sqrt{2}, -\sqrt{2}\), visualize the terminal side stretching from the origin, intersecting the fourth quadrant at this point.

This concept is essential to determine the correct calculation of the angle measure, such as knowing that \(315^\circ\) represents an angle in the fourth quadrant with the terminal side passing through the given point.