Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.
5x - 2y + 4 = 0, 3x + 5y = 6

Key Concepts

Slope of a Line

The slope of a line in a two-dimensional Cartesian coordinate system is a measure of its steepness, typically represented as 'm' in the slope-intercept form y = mx + b. It is calculated as the change in y divided by the change in x (rise over run). For two lines, their slopes are essential for determining the angle between them, as the angle can be derived from the difference in their slopes.

Tangent of an Angle

In trigonometry, the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. When finding the angle between two lines, the tangent of the angle can be calculated using the formula: tan(θ) = |(m1 - m2) / (1 + m1*m2)|, where m1 and m2 are the slopes of the two lines. This relationship is crucial for determining the acute angle between the lines.

Acute Angle

An acute angle is defined as an angle that measures less than 90 degrees. In the context of finding the angle between two lines, it is important to ensure that the calculated angle is acute, as the tangent function can yield both acute and obtuse angles. Rounding the result to the nearest tenth of a degree helps in providing a precise and practical answer for applications in geometry and trigonometry.