Problem 108 - Precalculus

Question

Find the equation of the line with positive slope that passes through the point \((a, 0)\) and makes an acute angle \(\theta\) with the \(x\) -axis. The equation of the line will be in terms of \(x\) \(a,\) and a trigonometric function of \(\theta .\) Assume \(a>0\)

Step-by-Step Solution

Answer

The line equation is \(y = \tan(\theta)x - \tan(\theta)a\).

Step 1: Identify Slope Using Trigonometry

Since the line makes an acute angle \(\theta\) with the \(x\)-axis and has a positive slope, the slope \(m\) of the line is equal to the tangent of the angle \(\theta\). Thus, we have \(m = \tan(\theta)\).

Step 2: Use Point-Slope Form of a Line

The point-slope form of a line equation is given by \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Since the line passes through the point \((a, 0)\), we substitute \(x_1 = a\), \(y_1 = 0\), and \(m = \tan(\theta)\) into the equation: \[ y - 0 = \tan(\theta)(x - a) \] This simplifies to \(y = \tan(\theta)x - \tan(\theta)a\).

Step 3: Express the Equation in Terms of x

Since the problem asks for the equation in terms of \(x\) only, and the line equation already expresses \(y\) as a function of \(x\), \(y = \tan(\theta)x - \tan(\theta)a\) is the final equation of the line. This is already in terms of \(x, a,\) and \(\tan(\theta)\).

Key Concepts

Understanding Slope

The concept of slope is fundamental in understanding linear equations. It represents how steep a line is. Imagine climbing a hill. The slope of the hill tells you how steep it is; a higher slope means a steeper climb. In mathematical terms, slope is defined as the change in the vertical direction (rise) divided by the change in the horizontal direction (run). When a line makes an angle with the x-axis, its slope can be found using trigonometric functions. Specifically, the slope is the tangent of the angle that the line makes with the x-axis. Hence, for an angle \(\theta\), the slope \(m\) is \(m = \tan(\theta)\). This explains why knowing the angle gives you information about the steepness of the line.

Utilizing Point-Slope Form

The point-slope form is a handy tool in algebra, particularly useful when you know a point on a line and its slope. The formula is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. In our exercise, we have the point \((a, 0)\), meaning \(x_1 = a\) and \(y_1 = 0\). The slope \(m\) is provided by the tangent of angle \(\theta\) as \(\tan(\theta)\). Substituting these into the point-slope form, we derive the equation of the line as follows:

• Start with the point-slope form: \(y - 0 = \tan(\theta)(x - a)\).
• Simplify it to \(y = \tan(\theta)x - \tan(\theta)a\).

This form is incredibly useful as it allows you to quickly create an equation as long as you know one point and the slope.

Connecting Trigonometric Functions

Trigonometric functions help connect angles with linear dimensions, like slopes. For lines, the tangent function stands out because it directly relates the angle a line makes with the x-axis to its slope.The line in question makes an acute angle \(\theta\) with the x-axis, meaning that \(\theta\) is less than \(90^\circ\). As angles become larger up until this point, the line becomes steeper and the value of \(\tan(\theta)\) increases too. This relationship is critical when working with linear equations and predicting how changes in \(\theta\) affect the line.In simple words, when you see an equation where slope is expressed as \(\tan(\theta)\), it is a direct application of trigonometric functions in geometry to provide a powerful method of controlling and understanding line behavior.