Problem 48 - Precalculus

Question

Find the angle \(\theta\) (in radians and degrees) between the lines. $$\begin{aligned} &2 x-y=2\\\ &4 x+3 y=24 \end{aligned}$$

Step-by-Step Solution

Answer

The angle between the given lines in radians is approximately 1.107 and in degrees it is approximately 63.43 degrees.

Step 1: Rewriting the equations to slope-intercept form

Rewrite the equations in the slope-intercept form. The first equation \(2x - y = 2\) can be rewritten as \(y = 2x - 2\) and the second equation \(4x + 3y = 24\) can be written as \(y = -\frac{4}{3}x + 8\). So, the slope of the first line, \(m1\), is 2 and the slope of the second line, \(m2\), is -4/3.

Step 2: Apply the formula to find the angle in radians and degrees

The formula for finding the angle, \(\theta\), between two lines with slopes \(m1\) and \(m2\) is \(\theta = atan\left(\left|\frac{m1 - m2}{1+m1 m2}\right|\right)\). Substituting the slopes calculated in the previous step, we get \(\theta = atan\left(\left|\frac{2 - (-4/3)}{1+2(-4/3)}\right|\right)\). This will give the angle in radians.

Step 3: Convert the angle from radians to degrees

After calculation, the angle in radians needs to be converted to degrees. The conversion can be done by using the formula \(degrees = radians \times \frac{180}{\pi}\). So, multiply the radians calculated earlier by \(\frac{180}{\pi}\) to get the angle in degrees.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is an easy way to express straight lines on a graph. It is given by the formula \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept. The slope \(m\) indicates how steep the line is, showing the rate of change in \(y\) with respect to \(x\).
Rewriting equations in slope-intercept form makes it straightforward to identify these important elements of a line.

• The first equation, \(2x - y = 2\), can be rewritten as \(y = 2x - 2\). This shows that \(m1 = 2\) with a y-intercept of \(-2\).
• The second equation, \(4x + 3y = 24\), can be rewritten as \(y = -\frac{4}{3}x + 8\). This shows that \(m2 = -\frac{4}{3}\) with a y-intercept of \(8\).

The ability to rewrite equations in this format helps when comparing lines and finding angles, as it gives direct access to the slopes needed for further calculations.

Angle Conversion

Once you have calculated an angle in radians, you may need to convert it to degrees. Angles can be measured in both radians and degrees, which are simply two different units for measuring angles. Understanding both is crucial in mathematics, especially when dealing with trigonometric calculations.
To convert an angle from radians to degrees, use the formula:

• \(degrees = radians \times \frac{180}{\pi}\)

For instance, if you have an angle of 1 radian, the conversion would be:

• \(degrees = 1 \times \frac{180}{\pi}\)

This yields approximately \(57.3\) degrees. Such conversions are often necessary because certain situations or interpretations of results might require angles in degrees, rather than in radians, which are the standard SI unit.

Trigonometric Formula

To find the angle between two lines, we use a trigonometric formula involving the slopes of these lines. The formula is:

• \(\theta = \tan^{-1}\left(\left|\frac{m1 - m2}{1 + m1 \cdot m2}\right|\right)\)

This formula helps us calculate \(\theta\), the angle formed between two intersecting lines. The process involves:

1. Subtracting the slope of one line from the other.
2. Dividing by \(1\) plus the product of the two slopes.
3. Taking the arctangent (\(\tan^{-1}\)) of the absolute value of this result.

This formula is derived from the concept of tangent slopes and gives the angle in radians initially. By using this method, you can effectively understand the relationship between two lines in terms of their angular separation. This formula provides a fundamental concept for geometry and vector analysis.