Problem 42 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line. $$2 x-6 y-12=0$$

Step-by-Step Solution

Answer

The inclination \(\theta\) of the line 2x - 6y -12 = 0 is approximately 0.3217 radians or 18.43 degrees when calculated.

Step 1: Writing the equation in the slope-intercept form

We begin by re-arranging the equation into the form y = mx + b. This gives us: 6y = 2x - 12, and then we divide through by 6 to get y = 1/3x - 2. Here, the coefficient of x is the slope m = 1/3.

Step 2: Finding the Angle θ in Radians

The inclination of the line can be found by taking the arctangent of the slope. Thus, ArcTan(m) = ArcTan(1/3) gives the value of the angle \(\theta\) in radians.

Step 3: Converting Radians to Degrees

Finally, we convert the radian measure to degrees by multiplying by \(180/\pi\) since there are \(180/\pi\) degrees in one radian. This gives us the measure of \(\theta\) in degrees.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is a way of writing linear equations so that you can easily identify the slope and y-intercept. It is expressed as \( y = mx + b \), where:

• \( m \) is the slope or steepness of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

To re-arrange an equation into this form, isolate \( y \) on one side of the equation. This involves moving terms around, and sometimes dividing by a coefficient. In our exercise, the equation \( 2x - 6y - 12 = 0 \) was transformed into \( y = \frac{1}{3}x - 2 \). This tells us:

• The slope \( m \) is \( \frac{1}{3} \), indicating a gentle upward incline.
• The line crosses the y-axis at \( -2 \).

Slope

The slope of a line shows how steep the line is and the direction it's heading. It is typically represented by \( m \), and is calculated as the change in y divided by the change in x between two points on the line \( (x_1, y_1) \) and \( (x_2, y_2) \). Mathematically, this is:\[m = \frac{y_2 - y_1}{x_2 - x_1}.\]If the slope is a positive number, the line ascends as it moves from left to right. If it's negative, the line descends. In the exercise equation, \( m = \frac{1}{3} \) shows a line that rises slowly.

• The larger the slope, the steeper the incline.
• A zero slope means the line is perfectly horizontal.
• An undefined slope (division by zero) indicates a vertical line.

Angle Conversion

Converting an angle between radians and degrees is often necessary in problem-solving. The inclination of a line is closely related to its slope, found using the arctangent function. We use:\[\theta = \arctan(m)\]In the exercise, \( \theta \) in radians was computed as \( \arctan(\frac{1}{3}) \). Radians are a way of measuring angles based on the radius of a circle. To convert from radians to degrees, use the formula:\[degrees = radians \times \frac{180}{\pi}.\]This conversion is critical because degrees are often more intuitive for interpreting angles in everyday scenarios. For example:

• A right angle is \( \frac{\pi}{2} \) radians, equivalent to 90 degrees.
• One radian is approximately 57.3 degrees.
• The full circle is \( 2\pi \) radians or 360 degrees.