Problem 13 - Precalculus

Question

Find the slope of the line with inclination \(\theta\). $$\theta=1.27 \text { radians }$$

Step-by-Step Solution

Answer

The slope of the line with inclination \( \theta = 1.27 \) radians is given by \( m = \tan(1.27) \).

Step 1: Identify the given angle

The angle is given as \( \theta = 1.27 \) radians. Understanding that this is in radians is key as it will directly affect the result.

Step 2: Compute the tangent of the given angle to find the slope

We know that the slope, m, of a line with an inclination angle \(\theta\) is given by \(m = \tan\theta\). Calculating the tangent of the given angle will yield the slope. So \(m = \tan(1.27)\).

Step 3: Calculate the slope value

Using a scientific calculator, determine the numerical value of the tangent from step 2 to get the slope. Remember to use the tangent function that handles radians since the given value is in radians.

Key Concepts

Understanding the Tangent Function

The tangent function is a fundamental concept in trigonometry. It relates an angle in a right triangle to the ratios of the lengths of the opposite side to the adjacent side. In a broader sense, in the unit circle, it is the length of the line segment from the origin to where the ray at angle \(\theta\) intersects the tangent line of the circle.- The tangent function is often used to find slopes in geometry because it represents the ratio that gives us this slope.- For angles measured in radians, the tangent function can be calculated using scientific calculators or through mathematical tables.It's crucial to understand that the tangent of an angle provides the measure needed to find the slope of a line with a given angle of inclination. This makes it an essential tool in solving geometric problems involving angles, such as the one in this exercise.

What are Radians?

Radians are a unit of angular measure used predominantly in mathematics. Unlike degrees, which are more common in everyday situations, radians offer a more natural measure for many mathematical applications.- One radian is the angle created when the length of the arc of a circle is equal to the circle's radius.- There are approximately \(2\pi\) radians in a full circle (or about 6.28318 radians).Using radians allows for a more seamless integration with other mathematical functions and equations, particularly in calculus and trigonometry. They become particularly relevant when working with trigonometric functions like tangent, which is crucial for finding the slope in this exercise. The consideration that the angle \(\theta = 1.27\) is in radians ensures calculations are precise and correct.

Exploring the Angle of Inclination

The angle of inclination of a line with respect to the positive direction of the x-axis is a geometric concept used to describe the steepness or tilt of the line.- Expressed in terms of an angle \(\theta\), typically measured from the x-axis upwards towards the line.- It ranges from \(0\text{ to }\pi\) radians, or \(0\text{ to }180\) degrees, where \(\pi/2\) is a vertical line.When solving problems like finding the slope, knowing this angle helps. This angle's tangent provides the slope directly using the formula \(m = \tan\theta\). In this specific exercise, having the angle of inclination in radians helps find the slope efficiently and accurately without conversion. This understanding of inclination is useful across various applications, from engineering to physics, wherever line tilt matters.