Problem 30 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line passing through the points. $$(12,8),(-4,-3)$$

Step-by-Step Solution

Answer

The inclination of the line is \(\frac{\pi}{4}\) radians or \(45^\circ\).

Step 1: Find the slope of the line

The slope of the line is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1, y_1) = (12, 8) \) and \( (x_2, y_2) = (-4, -3) \). Substituting these values gives \( m = \frac{-3 - 8}{-4 - 12} = 1 \).

Step 2: Calculate the angle in radians

The slope is equal to the tangent of the inclination angle \(\theta\). Therefore, we have \(\theta = \arctan(m) = \arctan(1) = \frac{\pi}{4} \) radians.

Step 3: Convert the angle to degrees

To convert from radians to degrees, use the conversion factor \(180^\circ/\pi \) radians. This gives the angle in degrees as \(\theta = \frac{180^\circ}{\pi} * \frac{\pi}{4} = 45^\circ\).

Key Concepts

Understanding the Slope Formula

The slope formula is an essential concept in understanding the inclination of a straight line. When given two points, the slope, denoted as \( m \), is calculated with the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two distinct points on the line. The numerator \( y_2 - y_1 \) represents the vertical change or the rise, while the denominator \( x_2 - x_1 \) represents the horizontal change or the run between these two points.

• If \( m > 0 \), the line rises as it moves from left to right.
• If \( m < 0 \), the line falls as it moves from left to right.
• If \( m = 0 \), the line is perfectly horizontal.
• An undefined \( m \) suggests a vertical line.

Understanding how these values change can help interpret the direction and steepness of a line. If given the points \( (12, 8) \) and \( (-4, -3) \), the slope is calculated as: \[ m = \frac{-3 - 8}{-4 - 12} = 1 \]This means the line rises diagonally with a one-to-one ratio of rise to run.

Angle Conversion: Radians and Degrees

Conversion between radians and degrees is a crucial skill in trigonometry and geometry. Radians and degrees are two different units for measuring angles.

• Radians are a natural unit of measure based on the radius of a circle.
• Degrees are more common in everyday use, divided into 360 parts for a full circle.

To convert from radians to degrees, you use the conversion factor \( \frac{180^\circ}{\pi} \). For example, if you have an angle \( \theta = \frac{\pi}{4} \), converting to degrees involves:\[ \theta_{degrees} = \frac{180^\circ}{\pi} \times \frac{\pi}{4} = 45^\circ \]This computation ensures you can effortlessly switch between these units based on the problem requirements while maintaining accuracy.

The Tangent of an Angle and Its Connection to Slope

The tangent of an angle is a central concept in trigonometry and is intricately linked to the slope of a line. In fact, the slope \( m \) of a line can be described as the tangent of its inclination angle \( \theta \). This relation can be expressed as:\[ m = \tan(\theta) \]When you know the slope \( m \), you can find the angle of inclination by taking the inverse tangent, or \( \arctan(m) \). It's vital to understand that:

• The tangent function links an angle to a ratio of two sides in a right triangle - opposite/adjacent.
• A positive slope signifies a line forming an acute angle with the horizontal axis.
• A negative slope implies the line forms an obtuse angle with the horizontal axis.

Given the slope \( m = 1 \), we determine the angle \( \theta \) using:\[ \theta = \arctan(1) = \frac{\pi}{4} \; \text{radians}, \; \text{or} \; 45^\circ \]Recognizing this relationship aids in visualizing how slopes correspond to specific angles of incline in geometric figures.