Problem 8 - Precalculus

Question

Find the slope of the line with inclination \(\theta\). $$\theta=\frac{2 \pi}{3} \text { radians }$$

Step-by-Step Solution

Answer

The slope of the line with inclination \(\frac{2 \pi}{3}\) radians is \(\sqrt{3}\).

Step 1: Identify the given inclination

The inclination, or angle, of the line is given as \(\theta = \frac{2 \pi}{3}\) radians. This is the angle the line makes with the positive x-axis.

Step 2: Apply the tangent relation

The slope m of the line is given by the tangent of the angle, represented by the formula m = tan(\(\theta\)).

Step 3: Calculate the slope

Calculate the slope by plugging \(\theta = \frac{2 \pi}{3}\) into the equation m = tan(\(\theta\)) which gives \(m = tan(\frac{2 \pi}{3})\) = \(\sqrt{3}\).

Key Concepts

Inclination Angle

The inclination angle of a line is a geometric concept that describes how steep or tilted a line is with respect to the horizontal axis. It is the angle formed between the positive x-axis and the line. This angle helps us understand the orientation of a line in a plane.

An inclination angle can range from 0 to 180 degrees when measured in a counterclockwise direction. For lines oriented downwards to the right, the angle will be less than 90 degrees, while lines that slope upwards to the left will have angles greater than 90 degrees. This angle is crucial in determining the slope of a line, as it directly influences the tangent function used to calculate the slope.

In mathematical problems, inclination angles are frequently expressed in radians or degrees. Remember that a slope and the inclination angle are connected through the trigonometric function known as tangent.

Radians

Radians are a unit for measuring angles, serving as an alternative to degrees. While degrees are familiar to most people—where a full circle measures 360 degrees—radians provide a more natural way to represent angles in mathematics due to their direct relationship with the properties of a circle.

One radian is the angle formed when the arc length of a circle is equal to the radius of the circle. As such, a complete circle measures about \(2\pi\) radians. The conversion from degrees to radians involves the formula: \( ext{radians} = ext{degrees} \times \frac{\pi}{180}\)

For example, an angle of \(\frac{2\pi}{3}\) radians equates to a little over 120 degrees. This understanding is crucial for applying trigonometric functions, such as finding the slope of a line given an inclination angle measured in radians.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides, but they also serve a broader mathematical role in relating angles to ratios. The primary trigonometric functions are sine, cosine, and tangent, each representing a ratio of different sides of a right triangle.

The tangent of an angle, in particular, is important for determining the slope of a line. The formula \(m = \tan(\theta)\) uses the tangent function to find the slope \(m\) based on the angle \(\theta\) that the line forms with the positive x-axis.

This is why knowing the tangent of angles, such as \(\frac{2\pi}{3}\), allows us to calculate the slope directly. In this example, \(\tan(\frac{2\pi}{3})\) results in a slope of \(-\sqrt{3}\). Mastery of trigonometric functions is essential for solving geometric problems involving angles and slopes.