Problem 50 - Precalculus

Question

Find the angle \(\theta\) (in radians and degrees) between the lines. $$\begin{aligned} &5 x+2 y=16\\\ &3 x-5 y=-1 \end{aligned}$$

Step-by-Step Solution

Answer

The angle between two lines is \(-\theta = -2.263 radians = -129.805 degrees\) or \(\theta = 0.879 radians = 50.195 degrees\).

Step 1: Convert equations into y = mx + b form

An equation is in the form of y = mx + b form, where m represents the slope of the line. It is possible to convert the given equations into this form by isolating y. The first equation, after the transformation, will be \(y = -\frac{5x}{2} + 8\) and the second - \(y = \frac{3x}{5} + \frac{1}{5}\)

Step 2: Identifying the slopes

In the equation of the line in slope-intercept form, the slope is the coefficient of x. So, the slope of the first line \(m1 = -\frac{5}{2}\) and the slope of the second one \(m2 = \frac{3}{5}\)

Step 3: Find the angle

We know both slopes, so now we may use the formula for the angle between two lines: \(\tan(\theta) = \frac{m1 - m2}{1 + m1*m2}\). Substituting the slopes of the lines into this formula we get \(\tan(\theta) = \frac{-\frac{5}{2} - \frac{3}{5}}{1 + -\frac{5}{2}*\frac{3}{5}}\). After simplifying this expression we get \(\tan(\theta) = -\frac{121}{46}\). Then find the arctangent (this could be done using a calculator) of resulting value to get the angle in radians. Finally, to convert radians to degrees, we multiply by 180 and divide by \(\pi\).

Step 4: Interpretation

A negative angle corresponds to a clockwise rotation. Since these lines are not parallel and we are interested in the smallest positive angle between them, the acute angle to consider is \(|-\theta|\) or \(-\theta + \pi\) radians and their respective degree analogues.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express linear equations simple and direct. This form shows how one variable, typically \(y\), changes with another, \(x\). The mathematical form is \(y = mx + b\), where:

• \(m\) is the slope, or steepness, of the line.
• \(b\) is the y-intercept, which is where the line crosses the y-axis.

Writing equations in this form allows us to easily identify the slope \(m\), which is crucial when comparing lines. For instance, in our problem, converting equations to this form helped us find the slopes \(-\frac{5}{2}\) and \(\frac{3}{5}\). These calculations are essential for further analysis, such as calculating angles between lines.

Trigonometric Functions

Trigonometric functions are fundamental in geometry, describing relationships between angle measures and side lengths in right triangles. The tangent function, \(\tan(\theta)\), is particularly useful for finding angles in various contexts, such as between two lines. We use the formula:

• \(\tan(\theta) = \frac{m_1 - m_2}{1 + m_1 \cdot m_2}\)

Here, \(m_1\) and \(m_2\) are the slopes of two lines. This formula helps find the tangent of the angle \(\theta\) between the lines. In our specific case, substituting the slopes, we dealt with \(\tan(\theta) = -\frac{121}{46}\). Calculators can be used to find \(\arctan\) (arctangent), providing an angle in radians.

Conversion Between Radians and Degrees

When dealing with angles, we often need to convert between radians and degrees. Radians measure angles based on the radius of a circle, while degrees divide a circle into 360 parts. This conversion is crucial since angles are commonly expressed in different units. The relationship is:

• \(1\ ext{radian} = \frac{180}{\pi}\ ext{degrees}\)
• Conversely, \(1\ ext{degree} = \frac{\pi}{180}\ ext{radians}\)

To convert radians to degrees, multiply by \(180\) and divide by \(\pi\). Thus, if you have an angle \(\theta\) in radians from \(\tan^{-1}( -\frac{121}{46})\), you convert it to degrees to understand it better in everyday terms. This process allows you to choose the most suitable measure for your needs.