Problem 10 - Precalculus with Limits

Question

In Exercises 5-12, find the slope of the line with inclination \(\theta\). \(\theta = \dfrac{5\pi}{6}\)radians

Step-by-Step Solution

Answer

The slope of the line with inclination \(1.27\) radians is approximately \(3.31\).

Step 1: Understanding the information given

The inclination of the line \(\theta\) is given as 1.27 radians.

Step 2: Applying the tangent function

We know that the slope of the line can be found by applying the tangent function to the given angle. Thus, we have to find the value of \(\tan(1.27)\).

Step 3: Calculating the slope using a calculator

Using a scientific calculator, calculate the value of \(\tan(1.27)\) to find the slope of the line.

Key Concepts

Inclination of a Line

The inclination of a line refers to the angle a line makes with the positive x-axis. This angle is fundamental in understanding the concept of the slope, as it provides a direct connection between the geometric angle and algebraic expression of a line's steepness. When looking at a graph, imagine the line rising or falling from left to right. If it moves upwards, the angle is positive, and if it moves downwards, the angle is negative.### Properties of Line Inclination

• It ranges from 0 to 180 degrees or from 0 to \(\pi\) radians.
• An angle of 0 or \(\pi\) radians means the line is horizontal and has no slope.
• An angle of \(\frac{\pi}{2}\) radians indicates a vertical line, where slope is undefined.

The inclination is crucial because it relates directly to the slope of the line through the tangent function.

Tangent Function

The tangent function is a trigonometric function that relates an angle of a right triangle to the ratios of two specific sides: the opposite and adjacent sides. In the context of a line, it represents the slope or steepness of the line by calculating the tangent of the line's inclination angle.### Tangent in Slope Calculation

• Slope \(m\) of a line is given by \(m = \tan(\theta)\), where \(\theta\) is the angle of inclination.
• The tangent function can be used with angles measured in radians for calculations in calculus and physics.

Since the tangent function can map any real number to a real number, it is particularly useful in determining the behavior of lines, whether they are steep or flat, or whether they slope upwards or downwards. In our exercise, calculating \(\tan(1.27)\) helps us find the slope with a given inclination in radians, providing a numerical representation of the line's incline.

Radians

Radians are an alternative measurement of angles, providing a natural way to express angles in terms of the arc length of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians are based on the radius of a circle.### Understanding Radians

• One full revolution around a circle (360 degrees) equals \(2\pi\) radians.
• A right angle is \(\frac{\pi}{2}\) radians, and a straight line is \(\pi\) radians.

In trigonometry and calculus, radians are often preferred because they simplify many formulas. They integrate seamlessly with the algebraic expressions involved in finding slopes. In our exercise, the inclination angle is given as 1.27 radians. This specific angle is already set for calculation without needing conversion from degrees, making it straightforward to apply in the tangent function.