Problem 44 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line. $$5 x+3 y=0$$

Step-by-Step Solution

Answer

After following the steps provided calculation should be done. The angle \( \theta \) in radians is found by the arctan function using a calculator. After finding the radian, we just multiply it by \( \frac{180}{\pi} \) using a calculator to transform it into degrees. So, the line's inclination can be represented in both radians and degrees.

Step 1: Convert equation to slope-intercept form

The equation can be rearranged as follows: \( y= -\frac{5}{3}x + 0 \). Therefore, the slope of the line (m) is -5/3 and the y-intercept (b) is 0.

Step 2: Find the angle of inclination in radians

The angle of inclination is given by \( \theta = \arctan(m) \), where m is the slope of the line. Plugging in our values, we get \( \theta = \arctan(-\frac{5}{3}) \). Use the \( \arctan \) function to calculate the angle in radians.

Step 3: Convert radians to degrees

To convert radians into degrees, multiply the radian measure by \( \frac{180}{\pi} \). Hence, \( \theta_{(degree)} = \theta_{(radian)} * \frac{180}{\pi} \) would give the result.

Key Concepts

Slope-Intercept Form

The slope-intercept form is an essential concept in algebra, especially when dealing with linear equations. It's expressed as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) refers to the y-intercept, the point where the line crosses the y-axis. This format makes it easy to identify these two crucial features of a line with just a glance.

In the equation from the exercise \( 5x + 3y = 0 \), converting this to the slope-intercept form involves rearranging it to solve for \( y \). This results in \( y = -\frac{5}{3}x + 0 \), revealing that the line has a slope \( m = -\frac{5}{3} \) and a y-intercept of 0. This means the line slopes downward as it moves from left to right on a graph.

Understanding this form is crucial because:

• It provides instant information about the direction and steepness of a line based on the slope \( m \).
• It allows for quick graphing of the line by identifying the y-intercept \( b \) as the starting point.
• It simplifies the process of finding the angle of inclination and other properties of the line.

Angle of Inclination

The angle of inclination of a line refers to the angle formed between the line and the positive direction of the x-axis. It's a measure of how steep a line is, with a larger angle suggesting a steeper line. This angle is often denoted by \( \theta \) and can be calculated using the slope \( m \) of the line.

The formula to find this angle in radians is \( \theta = \arctan(m) \), where \( \arctan \) is the inverse tangent function. In our problem, the slope \( m = -\frac{5}{3} \) leads us to calculate \( \theta = \arctan(-\frac{5}{3}) \). This approach allows for determining the angle using basic trigonometric principles.

Understanding the angle of inclination is helpful for:

• Visualizing the line’s orientation related to the x-axis.
• Comparing the steepness of different lines.
• Solving problems in geometry and physics where such inclinations matter.

Knowing how to find the angle of inclination is an essential tool in analyzing and interpreting the graphical representation of linear equations.

Radians to Degrees Conversion

Radians and degrees are two units used to measure angles. Radians are a more natural measure in mathematics, especially in trigonometry and calculus, whereas degrees are often used for everyday angle measurement.

The conversion between these two units is key when you want to translate your results into a more familiar format, or vice versa. The conversion formula is: \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \).

In our problem, once we've determined \( \theta \) in radians, we can then convert it into degrees using this formula. This gives a clearer understanding of the angle's size since degrees are a more common angle measurement in practical applications.

Why is understanding radians to degrees conversion important?

• It's crucial when communicating angle measurements in different contexts.
• It broadens scope in mathematical problems, allowing flexibility in your calculations.
• Helps bridge understanding between mathematical concepts and real-world applications.

Keen knowledge about these conversions aids in effectively switching between different mathematical and practical scenarios, ensuring accuracy in problem-solving.