Problem 43 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line. $$4 x+5 y-9=0$$

Step-by-Step Solution

Answer

The inclination of the given line is \(\arctan(-\frac{4}{5})\) in radians and \(\arctan(-\frac{4}{5}) * \frac{180}{\pi}\) in degrees.

Step 1: Transform equation into slope-intercept form

Rearrange the equation \(4x + 5y - 9 = 0\) to the form \(y = mx + c\). We get: \( y = -\frac{4}{5}x + \frac{9}{5} \). Thus, our slope \(m = -\frac{4}{5}\).

Step 2: Find the angle in radians

The inclination \(\theta\) in radians is found by taking the inverse tangent (arctangent) of the slope. So, \(\theta = \arctan (m)\). By plugging in the slope we found earlier, we obtain: \(\theta = \arctan(-\frac{4}{5})\).

Step 3: Find the angle in degrees

The inclination \(\theta\) in degrees is found by converting the radians to degrees. The formula is \(\theta_{\text{deg}} = \theta_{\text{rad}} * \frac{180}{\pi}\). By plugging our earlier answer, we get: \(\theta_{deg} = \arctan(-\frac{4}{5}) * \frac{180}{\pi}\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing the equation of a line. It is written as \( y = mx + c \), where:

• \( m \) is the slope of the line, which tells us how steep the line is.
• \( c \) is the y-intercept, where the line crosses the y-axis.

To transform any linear equation to the slope-intercept form, rearrange it so that \( y \) is isolated on one side of the equation. For example, if you start with the equation \( 4x + 5y - 9 = 0 \), you can rearrange it by solving for \( y \).
First, move \( 4x \) to the other side of the equation, giving you \( 5y = -4x + 9 \). Then, divide everything by 5 to solve for \( y \), resulting in \( y = -\frac{4}{5}x + \frac{9}{5} \). This gives you the slope-intercept form and identifies the slope \( m \) as \( -\frac{4}{5} \). Understanding this form helps in analyzing and graphing linear equations efficiently.

Inverse Tangent

The inverse tangent, also known as the arctangent, is a trigonometric function used to find angles. When you know the slope of a line, the inverse tangent can help you find the angle of inclination, \( \theta \), in radians.
The slope of a line \( m \) is related to the angle of inclination by the equation \( \theta = \arctan(m) \). This means you take the arctangent of the slope to find the angle in radians. for instance, for a slope \( m = -\frac{4}{5} \), you'd compute \( \theta = \arctan(-\frac{4}{5}) \).

• The inverse tangent function can handle both positive and negative slopes, making it versatile in finding the angle's direction.
• It's essential in determining how steep or shallow a line appears when graphed.

With tools like a calculator, finding \( \arctan \) becomes straightforward, providing results crucial for further conversions into degrees.

Radians to Degrees Conversion

Converting radians to degrees is a significant step to make angles more interpretable, as degrees are widely used in everyday contexts. The conversion is completed using the formula:

• \( \theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi} \)

This formula works because \( \pi \) radians is equivalent to 180 degrees, representing a half-circle. To convert an angle in radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \).
For instance, if the radian measure is \( \arctan(-\frac{4}{5}) \) from a previous computation, applying the conversion formula will give you the angle in degrees. Doing this allows for easier understanding of an angle's position or rotation, which is crucial in many fields such as navigation and engineering. Keeping a calculator handy is useful, as it simplifies and speeds up the conversion process.