Problem 9 - Precalculus

Question

Find the slope of the line with inclination \(\theta\). $$\theta=\frac{\pi}{3} \text { radians }$$

Step-by-Step Solution

Answer

The slope, m, of the line with inclination \( \frac{\pi}{3} \) radians equals \(\sqrt{3}\).

Step 1: Identify the formula

The relationship between the slope and the angle of inclination of a line in a plane is given by the formula:\n\[ m = \tan{\theta} \] \n where: \n m: slope of the line,\n \(\theta \): the angle of inclination in radians.

Step 2: Apply the given inclination into the formula

Substitute the given inclination angle \(\theta = \frac{\pi}{3}\) into the formula: \[\n m = \tan{\left(\frac{\pi}{3}\right)} \]

Step 3: Calculate the slope

Using a calculator or your knowledge of trigonometric values, calculate the value of the tangent of \(\frac{\pi}{3}\). This value is the slope (m) of the line.

Key Concepts

Angle of Inclination

When we talk about the angle of inclination of a line, we are referring to the angle that this line makes with the positive x-axis on the coordinate plane. This angle is usually measured in radians or degrees.
Here are some important facts to understand:

• The angle of inclination is always measured counterclockwise, starting from the x-axis.
• When a line rises from left to right, its angle of inclination falls between 0 and 90 degrees (or 0 and \( \frac{\pi}{2} \) radians).
• A horizontal line has an inclination of 0 degrees, while a vertical line has an inclination of 90 degrees (or \ \frac{\pi}{2} \ radians).

Understanding the angle of inclination is vital for figuring out the slope of a line. When you know the angle, you can use trigonometry to calculate the slope, which describes how steep the line is.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The most common trigonometric functions include sine, cosine, and tangent. These functions are fundamental in many areas of mathematics including geometry, calculus, and engineering.
The key trigonometric functions are:

• The sine function (\(\sin\theta\)) is the ratio of the opposite side to the hypotenuse in a right triangle.
• The cosine function (\(\cos\theta\)) is the ratio of the adjacent side to the hypotenuse.
• The tangent function (\(\tan\theta\)) is the ratio of the opposite side to the adjacent side.

For our work with slopes, we focus on the tangent function because it directly relates the angle of a line to its slope.

Tangent Function

The tangent function is essential when calculating the slope of a line given an angle of inclination. Specifically, the slope (\(m\)) of a line is equal to the tangent of its angle of inclination. This can be expressed with the formula: \[ m = \tan\theta \]
Here is why the tangent function is important:

• It provides a direct calculation of the slope, which helps describe how steep a line is relative to the x-axis.
• The tangent function is periodic with a period of \(\pi\), meaning it repeats its values in this interval.
• It can take on any real number value, reflecting the fact that slopes can be positive, negative, or undefined (in the case of vertical lines).

For example, to find the slope of a line where the angle of inclination is \(\frac{\pi}{3}\) radians, you would calculate \( \tan\left(\frac{\pi}{3}\right) \), which equates to \(\sqrt{3}\). This tells us that the slope of the line is \(\sqrt{3}\). Understanding this function is crucial for comprehending how angles determine the steepness of lines.