Problem 16 - Precalculus

Question

Find the slope of the line with inclination \(\theta\). $$\boldsymbol{\theta}=2.88 \text { radians }$$

Step-by-Step Solution

Answer

To find the final value of the slope, compute \(\tan(2.88)\) using a calculator.

Step 1: Identify Given Values

In this problem, the inclination of the line, \(\theta\), is given as 2.88 radians.

Step 2: Calculation of Slope

The slope (m) of a line given the inclination \(\theta\) can be found using the following relation: \(m = \tan(\theta)\). Apply the tangent function to the given angle to calculate the slope.

Step 3: Apply the Tangent Function

To find the slope, input the provided angle into the tangent function. This would look like: \(m = \tan(2.88)\).

Key Concepts

Inclination of a Line

The inclination of a line is a fundamental concept in geometry and trigonometry. It's essentially the angle that a line forms with the positive direction of the x-axis.
In simpler terms, think about it as the tilt of the line as you look at it from left to right.

• If a line rises from left to right, it has a positive inclination.
• If it declines, the inclination is negative.
• If a line is perfectly horizontal, the inclination is zero.

Knowing the inclination helps us describe both the direction and steepness of a line, which is useful in various math problems. In the context of coordinate geometry, the inclination is often expressed in terms of an angle, and this angle is measured in radians or degrees depending on the context.

Radians

Radians are a way of measuring angles and are often used in trigonometry. Unlike degrees, which divide a circle into 360 parts, radians divide it based on the radius of the circle. One whole circle equals about 6.283 radians, which is also known as 2π radians. Using radians has several advantages, especially in higher mathematics, because they are related directly to the properties of circles. For example:

• A quarter circle is π/2 radians (approximately 1.57).
• A half circle is π radians (approximately 3.14).
• A full circle is 2π radians (approximately 6.28).

When we say that a line has an inclination of 2.88 radians, we're specifying that angle with respect to the x-axis. Converting radians to degrees can be done using the relation: 1 radian equals approximately 57.29 degrees.

Tangent Function

The tangent function is a trigonometric function that helps relate angles to sides of a right triangle. In the context of lines, the tangent function helps us find the slope (or steepness) of the line.The slope of a line is given by \(m = \tan(\theta)\), where \(\theta\) is the inclination of the line.

• If \(\theta\) is known, you can find the slope by calculating \(\tan(\theta)\).
• The tangent of angle \((2.88)\) indicates the rate of change between the rise and run of the line.
• This relationship helps in plotting lines and finding intercepts on a coordinate grid.

The tangent function is crucial when dealing with slope because it simplifies the calculation of a line's steepness when the inclination is known. Understanding and using the tangent function makes dealing with angles and slopes smooth and straightforward.