Problem 33 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line passing through the points. $$\left(\frac{1}{4}, \frac{3}{2}\right),\left(\frac{1}{3}, \frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The inclination of the line passing through the points \((1/4, 3/2)\) and \((1/3, 1/2)\) in radians is approximately -1.107rad and in degrees is approximately -63.43°.

Step 1: Determine the slope of the line.

To determine the slope of the line, we utilize the formula for the slope of a line, given as \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the coordinates of the given points, we get \( m = \frac{\frac{1}{2} - \frac{3}{2}}{\frac{1}{3} - \frac{1}{4}}\). This simplifies to \(m = -2\).

Step 2: Calculate the angle of inclination.

The tangent of the angle of inclination of the line is equal to the slope of the line (This comes from the fact that Tangent of an angle in a right-angle triangle is the ratio of the opposite side to the adjacent side). Therefore, we can say tan(\(\theta\)) = m. Thus, the angle of inclination \(theta = tan^{-1} (-2)\). In order to calculate theta, we take the inverse tangent (also referred to as arctangent) of -2.

Step 3: Convert angles

Since we need the angle in both radians and degrees, we convert as required. Using a calculator, we find the radian angle is approximately -1.107 rad. To convert this to degrees, we use the conversion factor \( 1 rad \approx 57.296°\). Therefore, the angle in degrees would be approximately -63.43°.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in mathematics that describes the line's steepness and direction. It can be found using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where

• \( m \) is the slope
• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two distinct points on the line.

If the slope is positive, the line inclines upwards as it moves from left to right. Conversely, if it's negative, the line declines. A zero slope indicates a horizontal line, whereas an undefined slope is a vertical line. In our exercise, the slope is \(-2\), indicating a steep, declining line.

Trigonometric Functions

Trigonometric functions play a crucial role in understanding angles and their relationships to right triangles. The primary functions include sine, cosine, and tangent, among others. These functions relate an angle to ratios of sides in a triangle:

• Sine (sin): ratio of the opposite side to the hypotenuse.
• Cosine (cos): ratio of the adjacent side to the hypotenuse.
• Tangent (tan): ratio of the opposite side to the adjacent side.

For lines on a plane, the tangent of an angle formed with the horizontal can express the slope of that line. Therefore, understanding tangent is crucial for problems involving inclinations, as we see in this exercise.

Angle Conversion

Angles can be expressed in degrees or radians. Degrees are a more common unit, dividing a circle into 360 parts. Radians offer a direct relationship to the radius of a circle, with \(2\pi\) radians equivalent to a full circle.

• To convert radians to degrees, use the formula: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \)
• To convert degrees to radians, use: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)

For example, in the exercise, converting \(-1.107\) radians yields approximately \(-63.43^\circ\). These conversions are essential for interchanging angle measurements, especially when using a calculator that provides outputs in radians.

Inverse Trigonometric Functions

Inverse trigonometric functions reverse the process of trigonometric functions, returning an angle from a given trigonometric ratio. The principal inverse functions are:

• Arcsine (\( \sin^{-1} \)): inverse of sine.
• Arccosine (\( \cos^{-1} \)): inverse of cosine.
• Arctangent (\( \tan^{-1} \)): inverse of tangent.

In the calculation of a line's inclination, the arctangent is typically used: \( \theta = \tan^{-1}(m) \) where \( m \) is the slope. This function helps determine the angle when the ratio, provided by the slope in the context of lines, is known. In our example, \( \tan^{-1}(-2) \) calculates to approximately \(-1.107\) radians.