Problem 31 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line passing through the points. $$(-2,20),(10,0)$$

Step-by-Step Solution

Answer

The inclination \(\theta\) of the line passing through the points (-2,20) and (10,0) is given by \( \theta = \arctan(-\frac{5}{3}) \) in radians, and after converting, is equal to \(\theta (\frac{180}{\pi})\) in degrees.

Step 1: Calculate the slope \(m\)

First, use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) to find the slope of the line. Insert the provided points, so that \( (x_1, y_1) = (-2, 20) \) and \( (x_2, y_2) = (10, 0) \), which yields \( m = \frac{0 - 20}{10 - (-2)} = \frac{-20}{12} = -\frac{5}{3} \)

Step 2: Calculate the angle \(\theta\) in radians

Use the inverse tangent (or arctan) function to find the angle \(\theta\). \(\theta = \arctan(m)\). Since the slope \(m = -\frac{5}{3}\), it follows that \(\theta = \arctan(-\frac{5}{3})\). After calculating the value, ensure it is in the right quadrant depending on the direction of the line.

Step 3: Convert \(\theta\) to degrees

To convert \(\theta\) from radians to degrees, the expression \(\theta (\frac{180}{\pi})\) is used. Therefore, convert the value obtained in step 2 into degrees using this expression.

Key Concepts

Slope of a Line

The concept of the slope of a line is essential for understanding its steepness and direction. To calculate the slope \( m \) of a line, we use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula represents the vertical change (rise) over the horizontal change (run) between two points on the line.

• For example, given two points \((-2,20)\) and \((10,0)\), the slope \( m = \frac{0 - 20}{10 - (-2)} = -\frac{5}{3} \).
• The negative sign indicates a downward direction from left to right, while the magnitude \( \frac{5}{3} \) indicates the steepness.

Understanding the slope helps in visualizing the line on a graph and predicting how it behaves as we move along its path. The greater the absolute value of the slope, the steeper the line.

Angle Conversion

Converting angles between radians and degrees is a common task in mathematics, especially in trigonometry. Radians and degrees are two units for measuring angles.

• Degrees are based on dividing a circle into 360 parts, where a full circle is 360 degrees.
• Radians measure the angle based on the radius of a circle, with a full circle being \( 2\pi \) radians.

To convert radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \). For example, if an angle \( \theta \) is found to be in radians, multiplying \( \theta \) by \( \frac{180}{\pi} \) will give its equivalent in degrees. This is particularly useful in practical applications and when interpreting results in different contexts.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics for relating angles to side lengths in triangles and for analyzing periodic phenomena. These functions include sine, cosine, and tangent, which correspond to ratios within a right triangle.

• The tangent function, in particular, is used to relate an angle to the slope of a line: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• It's crucial in calculating the inclination of a line, as it directly connects the angle with the slope.

Understanding these functions allows for deeper insights into geometric properties and is essential for solving various mathematical problems. The tangent function helps us derive angles from slopes, aiding in interpretations of incline or declination.

Inverse Tangent Function

The inverse tangent function, also known as arctan or \( \tan^{-1} \), is used to determine the angle whose tangent is a given number. This function is particularly useful when you know the slope of a line and want to find the angle of inclination.

• For a slope \( m \), the angle \( \theta = \arctan(m) \).
• In our example, since the slope \( m = -\frac{5}{3} \), \( \theta = \arctan(-\frac{5}{3}) \) gives us the angle of inclination.

This angle is measured in radians and can be converted to degrees if necessary. It is important to keep the direction of the line in mind, as it affects the angle's quadrant. The inverse tangent function is practical for identifying angles in real-world applications, aiding in tasks like navigation and engineering design.