Problem 20 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line with slope \(m\). $$m=2$$

Step-by-Step Solution

Answer

The angle \(\theta\), in radians, is \(tan^{-1}(2)\) and in degrees, it is \(tan^{-1}(2) \times \frac{180}{\pi}\).

Step 1: Find Angle in Radians

We know that \(m= tan\, \theta\). Given that \(m=2\), we can solve for \(\theta\) by taking the inverse tangent. So, the calculation will be \(\theta = tan^{-1}(m)\) or \(\theta = tan^{-1}(2)\).

Step 2: Conversion to Degrees

The result from step 1 will be in radians. To convert this to degrees, we multiply by \(\frac{180}{\pi}\). So, the conversion will be as follows: \(\theta(degree) = \theta (radians) \times \frac{180}{\pi}\).

Step 3: Calculation

Plugging the given \(m=2\) into these equations, we get \(\theta(tan^{-1}(2))\) in radians and \(\theta(tan^{-1}(2) \times \frac{180}{\pi})\) in degrees.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in mathematics, representing the steepness or incline of the line. It is often denoted by the letter "m". The slope is calculated as the ratio of the vertical change to the horizontal change between two distinct points on a line. In simpler terms, it can be seen as the 'rise over run'. This is given by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.

The slope is crucial because it provides insight into the direction and angle of a line:

• If the slope is positive, the line inclines upwards from left to right.
• If the slope is negative, the line declines downwards from left to right.
• A zero slope indicates a horizontal line.
• An undefined slope corresponds to a vertical line.

Understanding the slope helps in determining other properties like angles and intercepts of the line. This is especially important in coordinate geometry and calculus.

Inverse Tangent

The inverse tangent, also known as arctan or \( an^{-1} \), is the operation used to find the angle whose tangent is a given number. In mathematical terms, if \( an \theta = m \), then \( heta = an^{-1}(m) \). This function is essential for finding angles when given the slope of a line.

The inverse tangent gives angles in radians, which is the standard unit of angular measure used in many areas of mathematics:

• It can cover values from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
• In terms of degrees, this range is from \(-90\degree\) to \(90\degree\).

When you need to determine the inclination angle of a line, you apply the inverse tangent to the slope. For example, in the original exercise, you would calculate \( \theta = \tan^{-1}(2) \) to find the angle in radians.

Angle Conversion

Converting angles between radians and degrees is a common requirement in mathematics, as both units are used in different contexts. Radians are based on the radius of a circle, while degrees divide a circle into 360 parts.

To convert an angle from radians to degrees, you use the conversion factor \( \frac{180}{\pi} \). This formula is derived from the fact that \(180°\) is equivalent to \(\pi\) radians. Hence:

• For an angle \( \theta \) in radians, multiply by \( \frac{180}{\pi} \) to get degrees.
• Conversely, to convert from degrees to radians, multiply by \( \frac{\pi}{180} \).

In practice, if you have an angle \( \theta = \tan^{-1}(2) \) in radians, you calculate the degree equivalent with \( \theta(degree) = \theta(radians) \times \frac{180}{\pi} \). This approach allows you to work flexibly between the two measurements depending on your needs.