Problem 163 - Precalculus

Question

The line \(y=\frac{3}{5} x\) passes through the origin in the \(x, y\) -plane. What is the measure of the angle that the line makes with the positive \(x\) -axis?

Step-by-Step Solution

Answer

The angle is approximately 30.96 degrees.

Step 1: Understand the slope

The line is given by the equation \(y=\frac{3}{5}x\), where \(\frac{3}{5}\) is the slope of the line. This means that for any unit change in \(x\), \(y\) changes by \(\frac{3}{5}\) units.

Step 2: Use the relation between slope and tangent

The slope of a line in the form \(y=mx\) is \(m\), which is equal to the tangent of the angle \(\theta\) that the line makes with the positive \(x\)-axis. Therefore, we have \(\tan(\theta) = \frac{3}{5}\).

Step 3: Find the angle using inverse tangent

To find \(\theta\), use the inverse tangent function: \(\theta = \tan^{-1}\left(\frac{3}{5}\right)\). Compute this angle using a calculator to determine its measure in degrees.

Step 4: Calculation of the angle

Use a calculator to evaluate \(\tan^{-1}\left(\frac{3}{5}\right)\), which results in approximately \(30.96\) degrees. This is the angle that the line makes with the positive \(x\)-axis.

Key Concepts

Slope of a Line

The slope of a line is a measure that indicates how steeply the line ascends or descends. The slope is essentially the ratio of the change in the vertical direction (rise) over the change in the horizontal direction (run) between two points on the line. In mathematical terms, the slope \( m \) of a line is defined using the formula: \[ m = \frac{\Delta y}{\Delta x} \]Where \( \Delta y \) is the change in \( y \) (vertical change), and \( \Delta x \) is the change in \( x \) (horizontal change).
For the line given by the equation \( y = \frac{3}{5}x \), the slope \( m \) is \( \frac{3}{5} \). This tells us that for every increase of 1 unit in the \( x \)-direction, the \( y \)-value increases by \( 0.6 \) units.

• Positive slope: Line rises as you move from left to right.
• Negative slope: Line falls as you move from left to right.

Tangent Function

The tangent function is one of the basic trigonometric functions that relates an angle of a right triangle to the ratio of two of its sides. Specifically, for an angle \( \theta \) in a right triangle, the tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side. In mathematical terms, it is represented as: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
When it comes to lines on a plane, the tangent of the angle \( \theta \) that a line makes with the positive \( x \)-axis is equal to the slope \( m \) of the line. This is crucial in understanding how the slope can be used to find the angle with respect to the \( x \)-axis. Thus, if \( \text{slope} = \frac{3}{5} \), then \( \tan(\theta) = \frac{3}{5} \).
The tangent function allows us to explore the relationship between linear equations and geometric angles.

Inverse Tangent Function

The inverse tangent function, often called \( \arctan \) or \( \tan^{-1} \), helps us determine the angle \( \theta \) when we know its tangent value. Essentially, it performs the opposite operation of the tangent function. If you have \( \tan(\theta) = \frac{3}{5} \), using the inverse tangent tells us \( \theta \) itself.
The formula to calculate the angle would be: \[ \theta = \tan^{-1}\left(\frac{3}{5}\right) \] To use the inverse tangent, you typically need a calculator that has this function available.

• In calculators, this function is usually found as \( \tan^{-1} \) or \( \arctan \).
• It gives the angle \( \theta \) in radians or degrees; make sure your calculator is set to the desired mode.

By using \( \arctan(\frac{3}{5}) \), you can find the precise angle \( \theta \), which is approximately \( 30.96 \) degrees.

Degrees Measure

Angles can be measured in various units, including degrees and radians. Degrees are the most common unit in everyday situations. There are 360 degrees in a full circle. Each degree can be further divided into minutes and seconds, but the primary unit of measurement is sufficient for many geometry problems.
When measuring angles in a coordinate plane, such as the angle a line makes with the \( x \)-axis, this is typically given in degrees. In our example, after using the inverse tangent, the angle was found to be approximately \( 30.96 \) degrees.

• A right angle is 90 degrees.
• A straight line makes a 180-degree angle.
• A full circle maps to 360 degrees.

Understanding degrees makes it simpler to visualize and comprehend angles, especially when working with topics like trigonometry and geometry.