Problem 29
Question
In Exercises 27-36, find the inclination \(\theta\) (in radians and degrees) of the line. \(3x - 3y + 1 = 0\)
Step-by-Step Solution
Verified Answer
The inclination \(\theta\) of the line \(3x - 3y + 1 = 0\) is approximately 0.785 radians or 45°.
1Step 1: Convert to Slope-Intercept Form
To find the inclination of the line, convert the equation into the slope-intercept form. The given equation is \(3x - 3y + 1 = 0\). Let's isolate 'y' on one side: \n\(y = x + \frac{1}{3}\). So, m (the slope) = 1.
2Step 2: Calculate Inclination in Radians
Substitute the slope into the formula for inclination, which is \(\theta = tan^{-1}(m)\). We have \(\theta = tan^{-1}(1)\). Using a calculator, we find \(\theta \approx 0.785\). So, in radians, the inclination is approximately 0.785.
3Step 3: Calculate Inclination in Degrees
Because the student is required also to find the angle in degrees, convert the radian measure to degrees. We multiply by \(\frac{180}{\pi}\) to do this. So the inclination in degrees is \(\theta = 0.785 * \frac{180}{\pi} \approx 45°\).
Key Concepts
Slope-Intercept FormRadians to Degrees ConversionTangent InverseSlope of a Line
Slope-Intercept Form
The slope-intercept form is a way of writing the equation of a straight line. It's written as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept, which is where the line crosses the y-axis.
The slope indicates how steep the line is. A larger slope means a steeper line. In our given equation \(3x - 3y + 1 = 0\), we need to rearrange it to find \(m\) and \(b\).
The slope indicates how steep the line is. A larger slope means a steeper line. In our given equation \(3x - 3y + 1 = 0\), we need to rearrange it to find \(m\) and \(b\).
- First, move the \(x\) term to the right side to isolate \(y\): \(3x - 3y = -1\).
- Next, solve for \(y\) by dividing the entire equation by \(-3\): \(y = x + \frac{1}{3}\).
Radians to Degrees Conversion
Radians and degrees are two units of measuring angles, and conversion between them can be done using a simple formula. One full circle is 360 degrees or \(2\pi\) radians.
To convert radians to degrees, you use the formula:
\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]
For instance, if you have an angle of \(0.785\) radians, as calculated from the tangent inverse in the problem, you multiply it by \(\frac{180}{\pi}\):
\[0.785 \times \frac{180}{\pi} \approx 45\] degrees.
This straightforward calculation allows us to switch between the two systems of measurement easily.
To convert radians to degrees, you use the formula:
\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]
For instance, if you have an angle of \(0.785\) radians, as calculated from the tangent inverse in the problem, you multiply it by \(\frac{180}{\pi}\):
\[0.785 \times \frac{180}{\pi} \approx 45\] degrees.
This straightforward calculation allows us to switch between the two systems of measurement easily.
Tangent Inverse
The tangent inverse, denoted as \(\tan^{-1}\), helps find the angle whose tangent is a given number. This function is crucial in finding the inclination of a line when you know its slope.
For any slope \(m\), the inclination \(\theta\) can be calculated as:
\[\theta = \tan^{-1}(m)\]
In our exercise, the slope \(m\) is 1. Hence, \(\theta = \tan^{-1}(1)\), which approximately evaluates to \(0.785\) radians.
This value represents the angle from the positive x-axis to the line, highlighting the angle's emphasis on direction and rotation.
For any slope \(m\), the inclination \(\theta\) can be calculated as:
\[\theta = \tan^{-1}(m)\]
In our exercise, the slope \(m\) is 1. Hence, \(\theta = \tan^{-1}(1)\), which approximately evaluates to \(0.785\) radians.
This value represents the angle from the positive x-axis to the line, highlighting the angle's emphasis on direction and rotation.
Slope of a Line
The slope of a line is a measure of how steep the line is. It is the ratio of the change in y to the change in x, often expressed as "rise over run."
Mathematically, the slope \(m\) can be calculated when you have a linear equation in the form \(y = mx + b\).
For the equation \(3x - 3y + 1 = 0\), after rewriting it in slope-intercept form as \(y = x + \frac{1}{3}\), the slope is clearly \(m = 1\).
This tells us the line rises 1 unit for every unit it moves horizontally, indicating a 45-degree incline in the standard xy-coordinate plane when considering the positive slope towards the upward direction.
Mathematically, the slope \(m\) can be calculated when you have a linear equation in the form \(y = mx + b\).
For the equation \(3x - 3y + 1 = 0\), after rewriting it in slope-intercept form as \(y = x + \frac{1}{3}\), the slope is clearly \(m = 1\).
This tells us the line rises 1 unit for every unit it moves horizontally, indicating a 45-degree incline in the standard xy-coordinate plane when considering the positive slope towards the upward direction.
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