Problem 21 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line with slope \(m\). $$m=\frac{3}{4}$$

Step-by-Step Solution

Answer

To compute the final answer, one needs a calculator handy to evaluate \(\tan^{-1}\) and convert radians to degrees. The exact results will depend on the calculator's precision.

Step 1: Calculate the inclination in radians

Using the provided slope \(\frac{3}{4}\), we set up this equation using the tangent function and its inverse: \(\tan(\theta) = m\) , giving us \(\theta= \tan^{-1}(m)\). Substituting \(\frac{3}{4}\) for \(m\), we get \(\theta=\tan^{-1}(\frac{3}{4})\)

Step 2: Compute the angle

To find the angle in radians, use a calculator to evaluate \(\tan^{-1}(\frac{3}{4})\)

Step 3: Convert radians to degrees

To convert from radians to degrees, multiply the radian result by \(\frac{180}{\pi}\). Let's denote the degree measure by \(D\). Then we have: \(D = \theta \times \frac{180}{\pi}\)

Key Concepts

Understanding the Concept of Slope

The slope of a line is a crucial concept in mathematics that helps you understand how steep a line is. Imagine you're hiking up a hill—you want to know how steep your path is. The slope tells you exactly that. It's the ratio of the vertical change to the horizontal change. Simply put:

• Vertical change is how far up or down you go.
• Horizontal change is how far left or right you go.

For our line, the slope is given as \(m = \frac{3}{4}\). This means that for every 4 units you move horizontally, you go up 3 units vertically. A higher slope means a steeper line, while a lower slope means a flatter line.
This key idea helps you find the line's inclination.

Exploring the Inverse Tangent

When finding the inclination of a line given its slope, the inverse tangent (also known as arctangent) is your go-to tool. The slope \(m\) relates to the tangent of the angle \(\theta\) as \(\tan(\theta) = m\). To find \(\theta\), we use the inverse function: \(\theta = \tan^{-1}(m)\).
For the slope \(\frac{3}{4}\), you calculate \(\theta = \tan^{-1}\left(\frac{3}{4}\right)\). This calculation gives you the angle \(\theta\) in radians. Using an inverse on a calculator helps you pinpoint exact angles. This step translates the slope's steepness into a tangible angle value. It's a handy technique for trigonometry problems!

Converting Angles: Radians and Degrees

Angles can be expressed in two main units: radians and degrees. Imagine a pizza cut into slices. Degrees tell you how many slices there are in 360° of pizza. Radians, however, are based on the circle's radius.

• Radians are often used in calculus and physics.
• Degrees are more familiar and intuitive for many people.

Converting between these units is essential depending on the task. It’s done using the conversion: \(1\) radian \( = \frac{180}{\pi}\) degrees. This allows you to translate between these two units seamlessly.

Switching from Radians to Degrees

After finding the angle \(\theta\) in radians using \(\tan^{-1}\left(\frac{3}{4}\right)\), sometimes you need to report the angle in degrees. This involves a simple conversion. Multiply the radian measure by \(\frac{180}{\pi}\) to transform it into degrees.
For example, if \(\theta\) were found to be \(x\) radians, the degree measure \(D\) would be \(D = x \times \frac{180}{\pi}\). This step ensures that the angle is expressed in a format that's often easier for people to understand and use in everyday situations. It's a practical skill for math and beyond, helping make complex concepts accessible.