Problem 27 - Precalculus with Limits

Question

In Exercises 27-36, find the inclination \(\theta\) (in radians and degrees) of the line. \(2x + 2y -5 = 0\)

Step-by-Step Solution

Answer

The inclination of the line \(2x + 2y -5 = 0\) is \(-\pi/4\) radians or \(-45\) degrees.

Step 1: Identify A and B in the equation

The equation given is \(2x + 2y - 5 = 0\), where A is 2 and B is also 2.

Step 2: Calculate angle in radians using arctan function

Since \(A = B = 2\), the tangent of the inclination angle is \(-A/B = -2/2 = -1\). Then, the inclination angle is the arctangent of -1, i.e., \(\theta = \arctan(-1)\). Using the unit circle, we find that \(\theta = -\pi/4\) radians.

Step 3: Convert radians to degrees

To convert the inclination angle from radians to degrees, use the relationship \(\pi\) radians = 180 degrees. Thus, \(\theta = -\pi/4 \times (180/\pi) = -45\) degrees.

Key Concepts

Radian to Degree Conversion

When dealing with angles, converting between radians and degrees is a common task. Radians and degrees are two units used to measure angles. To convert an angle from radians to degrees, you can use the conversion factor:

• \[ \pi \text{ radians} = 180^{\circ} \]

So, to convert radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \). Conversely, to convert degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \).
This conversion is essential when you encounter angles given in different units, especially in trigonometry problems where one is in radians, and results might be needed in degrees for better interpretation.

Arctangent Function

The arctangent function, often written as \( \arctan(x) \), is the inverse of the tangent function. It returns the angle whose tangent is \( x \).
For example, if \( \tan(\theta) = x \), then \( \theta = \arctan(x) \). The output of \( \arctan(x) \) is typically expressed in radians and falls within the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\).
In solving linear equations to find the inclination, using \( \arctan \) helps find the angle of a line relative to the x-axis, especially when dealing with slopes. For example, \( \arctan(-1) \) gives \(-\frac{\pi}{4} \) radians, which corresponds to an angle of \(-45^{\circ} \) after conversion to degrees.

Linear Equations

Linear equations are algebraic equations of the first degree, meaning they involve no exponents greater than one. The general form is \( Ax + By + C = 0 \). Here, \( A \) and \( B \) are coefficients, and \( C \) is a constant. These equations represent straight lines on a coordinate plane.
The slope of the line, given by \(-\frac{A}{B}\), indicates its steepness and direction. Positive slopes rise to the right, while negative slopes fall to the right.
For example, the equation \( 2x + 2y - 5 = 0 \) has a slope of \(-1\), leading to an inclination angle that can be derived using trigonometric functions such as the arctangent.