Question

Angle Between Two Lines Find the angle of intersection between line \(L_{1}\) having a slope of 3 and line \(L_{2}\) having a slope of -2.

Short answer

\( \theta = \frac{\pi}{4} \) radians or 45 degrees

Step-by-step solution

Recall the formula for the angle between two lines

The formula to find the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is \( \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right| \) where \( |x| \) denotes the absolute value of \( x \) and \( \tan \) is the tangent function.

Substitute the slopes into the formula

Substitute the given slopes into the formula: \( m_1 = 3 \) and \( m_2 = -2 \) to get \( \tan(\theta) = \left| \frac{3 - (-2)}{1 + (3)(-2)} \right| = \left| \frac{5}{1 - 6} \right| = \left| \frac{5}{-5} \right| = 1 \) since absolute value negates the negative sign.

Find the angle from the tangent value

To find \( \theta \) when \( \tan(\theta) = 1 \) we take the inverse tangent function \( \arctan \) of 1 to get \( \theta = \arctan(1) \) which is \( \frac{\pi}{4} \) or 45 degrees, because \( \tan(\frac{\pi}{4}) = 1 \) in the first quadrant where angles are positive.

Key concepts

Slope of a Line

The slope of a line is a measure of its steepness and direction. Mathematically, it's defined as the change in the y-coordinate (rise) over the change in the x-coordinate (run) between two distinct points on the line. If you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line, the slope \( m \) is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

In the context of finding the angle between two lines, the slopes offer a way to gauge their relative orientations. A positive slope indicates that the line rises from left to right, while a negative slope means that the line falls as it moves from left to right. When a line is horizontal, its slope is zero, and when it's vertical, its slope is undefined because you would be dividing by zero.

For the lines \( L_{1} \) with a slope of 3 and \( L_{2} \) with a slope of -2, the slope tells us that \( L_{1} \) rises three units vertically for every one unit horizontal change, while \( L_{2} \) falls two units vertically for the same horizontal change. This difference in slopes is key in calculating the angle of intersection between the two lines.

Tangent Function

The tangent function is a fundamental concept in trigonometry, relating the angles of a right triangle to the ratios of its sides. It is the ratio of the length of the opposite side to the length of the adjacent side of a right-angled triangle for a given angle \( \theta \). Mathematically, this is expressed as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

The tangent function is periodic with a period of \( \pi \) radians (or 180 degrees), which means it repeats its values every \( \pi \) radians. It is undefined for angles where the adjacent side is 0, namely at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \) (or 90 and 270 degrees), as this would require division by zero.

In the context of finding the angle between two lines, the tangent function comes into play as it is linked to the slope of the lines. Specifically, if you know the slopes of two intersecting lines, you can use the tangent of the angle between them to determine the measure of that angle.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find an angle given the value of a trigonometric function. For instance, the inverse tangent function, \( \arctan(x) \), returns the angle \( \theta \) whose tangent is \( x \). This is crucial when we deal with problems such as finding the angle between two lines because once we have the value of the tangent, we can use the inverse tangent function to find the measure of the angle.

While the tangent function repeats every \( \pi \) radians, its inverse, the \( \arctan \), typically ranges from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) (or -90 to 90 degrees). This ensures that every value of the tangent function corresponds to a single angle within that range, giving a unique solution.

In the exercise provided, once we determined that \( \tan(\theta) = 1 \), we apply the \( \arctan \) function to find that \( \theta = \arctan(1) = \frac{\pi}{4} \) or 45 degrees. This step is critical, as it bridges the value of a trigonometric function to an actual angle measurement, which can be visualized and understood in a geometric context.