Problem 26 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line passing through the points. $$(1,2 \sqrt{3}),(0, \sqrt{3})$$

Step-by-Step Solution

Answer

The inclination of the line passing through the given points is \(-\frac{\pi}{3}\) radians or \(-60^{\circ}\)

Step 1: Calculate the slope

The slope of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by: \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\), where \(m\) is the slope. In this case, using the given points \((1,2 \sqrt{3})\) and \((0, \sqrt{3})\), the slope \(m\) becomes: \(m = \frac{{\sqrt{3} - 2 \sqrt{3}}}{{0 - 1}} = - \sqrt{3}\)

Step 2: Find the angle in radians

Since the slope of the line is equivalent to the tangent of the angle of inclination, we can find the angle \(θ\) by using the arctan function: \(θ = \text{arctan}(m)\). Thus: \(θ = \text{arctan}(-\sqrt{3}) = -\frac{\pi}{3}\) radians

Step 3: Convert the angle to degrees

To convert an angle from radians to degrees, the appropriate conversion is to multiply the angle in radians by \(\frac{180}{\pi}\). Doing this gives: \(θ = -\frac{\pi}{3} \times \frac{180}{\pi} = -60^{\circ}\)

Key Concepts

Understanding Slope Calculation

The slope of a line represents its steepness and direction. It is calculated using two points on the line. If you know the coordinates of these points, finding the slope becomes straightforward. Let’s use some notation: suppose you have points \( (x_1, y_1) \) and \( (x_2, y_2) \). The formula for the slope \( m \) is given by:

• \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \)

This equation tells us that the slope is the change in the y-coordinates divided by the change in the x-coordinates. A positive slope means the line is going upwards as you move from left to right, while a negative slope means it’s going downwards. In our problem, the points provided are \( (1, 2\sqrt{3}) \) and \( (0, \sqrt{3}) \). By plugging these into the slope formula, we get:

• \( m = \frac{{\sqrt{3} - 2\sqrt{3}}}{{0 - 1}} = -\sqrt{3} \)

This tells us the line is steep, and it descends as you move to the right, with a slope of \( -\sqrt{3} \). Understanding this step is crucial for moving on to calculate the angle of inclination.

Converting Radial and Angular Measurements

When working with angles, it’s often necessary to convert between radians and degrees, especially when wanting to comprehend the inclination of a line in a more intuitive sense. Radians and degrees are just two different units for measuring angles, similar to how meters and feet measure length. To convert from radians to degrees, the equation is:

• \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \)

This conversion factor of \( \frac{180}{\pi} \) stems from the fact that \( 2\pi \) radians is equal to \( 360^{\circ} \). By using this conversion, we can turn radian estimates into degree measures. In our specific problem, we found the angle \( \theta \) (in radians) using the arctan function, which was \( -\frac{\pi}{3} \). Multiplying by the conversion factor:

• \( \theta = -\frac{\pi}{3} \times \frac{180}{\pi} = -60^{\circ} \)

This shows us the line is inclined at an angle of \( -60^{\circ} \), where a negative sign indicates it is measured clockwise from the positive x-axis. This measurement helps visualize how the line tilts in relation to a flat, horizontal surface.

Exploring the Arctan Function

The arctan (or inverse tangent) function is a mathematical operation used to find an angle when the slope of a line is known. \( \text{arctan} \) essentially asks, "What angle has a tangent equal to this value?" It is especially useful in trigonometry when dealing with right triangles and the unit circle. Let’s delve into how it works.

• The arctan of a slope \( m \) gives the angle \( \theta \) such that \( \tan(\theta) = m \).

In simpler terms, if you know the slope, you can find the angle of inclination using this function. For our line with slope \( -\sqrt{3} \), using the arctan function yields:

• \( \theta = \text{arctan}(-\sqrt{3}) = -\frac{\pi}{3} \)

This calculation gives us the angle in radians. An important note is that arctan will return values from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), corresponding to the range of potential angles in standard position on a coordinate plane. Understanding the arctan function is fundamental in linking slopes to their corresponding angles.