Problem 45 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

A line with slope \(m\) passes through the origin. An angle \(\theta\) in standard position has a terminal side that coincides with the line. Use a trigonometric function to relate the slope of the line to the angle.

Step-by-Step Solution

Answer

The slope of the line, \(m\), can be related to the angle \(\theta\) with the equation \(m = \tan(\theta)\).

Step 1: Visualize the Problem

Draw a line through the origin with slope \(m\). This creates a right triangle between the x-axis, the line, and the line perpendicular to the x-axis that goes through the point where the line intersects with the unit circle. This visualization will help to understand the problem more intuitively.

Step 2: Recognize Trigonometric Relationship

Realize that the slope of a line is defined as the tangent of the angle it makes with the x-axis. As our line is also intersecting the origin, the slope of the line is equivalent to the ratio of the y-coordinate to the x-coordinate for any given point on the line.

Step 3: Relate Slope to Tangent

By the definition of tangent in a right triangle (which is opposite side/adjacent side), and given that slope of the line is \(m\), can be related to the angle \(\theta\) by the equation \(m = \tan(\theta)\). This is the relation needed.

Key Concepts

Understanding the Slope of a Line

A slope is essentially a measure that tells us how steep a line is. When a line passes through the origin, it can either rise or fall as it moves from left to right. The slope of this line is a crucial concept because it provides a numerical value that describes this incline or decline.

Slope can be determined by the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. For a line that passes through the origin, the slope is simply the change in the y-coordinate divided by the change in the x-coordinate for any point on the line other than the origin itself.

This is often expressed with the formula:

• Let's consider a line and a point (0,0) and (x,y).
• The slope (m) is calculated as: \[m = \frac{y}{x}\]

This calculation tells you how much y changes as x changes. A positive slope indicates the line inclines upward, while a negative slope indicates it declines downward. In simpler terms, the line's slope gives you a picture of how fast the line rises or drops.

Connecting Slope and Tangent

The tangent of an angle in a right triangle is a fundamental concept in trigonometry. It is defined as the ratio of the length of the opposite side to the length of the adjacent side. This concept becomes very useful when dealing with the slope of a line.

Consider a line with slope \(m\) passing through the origin. The angle \(\theta\) that this line makes with the positive direction of the x-axis is linked to the slope by the trigonometric function tangent.

In mathematical terms, the slope of the line can be represented as:

• \(m = \frac{y}{x}\)
• For a line intersection at the origin, if \(y\) is the vertical distance and \(x\) is the horizontal distance, then by definition of tangent: \[\tan(\theta) = \frac{y}{x}\]
• Thus, the slope of the line is equivalent to the tangent of the angle \(\theta\):\[m = \tan(\theta)\]

By relating these two concepts, you can see that tangent not only measures how far a point on the line is vertically from the x-axis, but also explains how really that affects the slope of a line.

Exploring the Angle in Standard Position

In trigonometry, angles can be described in what is referred to as "standard position." This means that the angle's vertex is placed at the origin on a Cartesian coordinate system, and its initial side lies along the positive x-axis. The angle's terminal side is where the other ray extends from the origin.

For angles where the terminal side coincides with the line of a specific slope, understanding this configuration helps relate the angle to trigonometric functions.

• The angle \(\theta\) is measured counterclockwise from the positive x-axis.
• In relation to a line passing through origin with a given slope (representing tangent), \(\theta\) effectively describes the steepness or incline direction of that line.
• This alignment is crucial, because the trigonometric function tangent of \(\theta\) provides the slope \(m\) of the line.

By following this setup of angles, we can easily connect it to trigonometric concepts such as slope and tangent. Whereas the angle defines how the line is oriented relative to the x-axis, knowing that the slope and tangent are equal provides a straightforward method to calculate and understand how a line behaves on a graph.