Problem 10 - Precalculus

Question

Find the slope of the line with inclination \(\theta\). $$\theta=\frac{5 \pi}{6} \text { radians }$$

Step-by-Step Solution

Answer

The slope of the line is -√3.

Step 1: Identify the Radian Value of the Inclination

It is given that \(\theta = \frac{5\pi}{6} \) radian. This is the angle of inclination for the straight line.

Step 2: Calculate the Tangent of the Angle

The slope of the line can be determined by the tangent of the angle of inclination. That is, Slope \( = \tan{ \theta }\); Hence we calculate \(\tan{ \frac{5\pi}{6} }\).

Step 3: Compute the Value

Using the fact that \(\tan{ \frac{5\pi}{6} } = -\sqrt{3}\), we find that the slope of the line is \(-\sqrt{3}\).

Key Concepts

Angle of Inclination

When talking about lines in geometry, the angle of inclination is the angle formed by the line and the positive x-axis of a coordinate plane. This angle is always measured upwards from the x-axis towards the line. The angle of inclination can range from 0 to 180 degrees. The concept is crucial in determining the characteristics of a line, such as its direction and steepness.

• If the angle is 0 degrees, the line is perfectly horizontal.
• An angle of 90 degrees indicates a vertical line.
• Angles greater than 90 degrees describe lines that decline rather than incline as they extend from left to right.

The angle of inclination, when expressed in radians, offers an opportunity for precise calculations and transformations, especially in trigonometry. For example, an angle of inclination of \(\frac{5\pi}{6}\) radians represents a line that angles downward as it moves to the right, depicting a slope (or steepness) that is negative.

Radians

Radians are a unit of angular measure used in many areas of mathematics. Unlike degrees, which split a full circle into 360 equal parts, radians divide it based on the circle's radius.

In simpler terms:

• A radian is the angle created when you take the radius of a circle and wrap it along the circle's edge.
• In a full circle, there are \(2\pi\) radians, which is equal to 360 degrees.

To convert between degrees and radians, remember:

• Degrees to radians: Multiply by \(\frac{\pi}{180}\).
• Radians to degrees: Multiply by \(\frac{180}{\pi}\).

Thus, \(\frac{5\pi}{6}\) radians can be converted into degrees, providing a handy way to visualize this angle when needed.

Tangent of an Angle

The tangent of an angle in trigonometry is a vital concept when working with triangles and angles on the coordinate plane. It represents the ratio of the opposite side to the adjacent side in a right-angled triangle.

Mathematically, it is expressed as:\[\tan{\theta} = \frac{\text{Opposite side}}{\text{Adjacent side}}\]In terms of a line on a coordinate plane, the tangent of the angle of inclination is equal to the slope of the line. This means that:

• The slope of a line = \(\tan{\theta}\).

Given \(\theta = \frac{5\pi}{6}\), we calculate \(\tan{\frac{5\pi}{6}} = -\sqrt{3}\). This result indicates that the slope of the line is \(-\sqrt{3}\), describing a downward trend as the line moves from left to right.