91Ó°ÊÓ

In the context of straight lines, the slope is a measure of how steep the line is. It tells us the rate at which one variable changes with respect to another. If you're moving along the x-axis, the slope indicates how much the y-axis will change. A positive slope rises from left to right, while a negative slope falls from left to right.
For example, if you have a slope of -2, it means that for every unit you move to the right along the x-axis, the line drops 2 units along the y-axis. In terms of angles, the tangent of an angle \( \theta \) formed with the x-axis gives the slope of the line. Therefore, if a line makes an angle \( \theta \) with the x-axis, its slope is \( \tan(\theta) \). The negative sign in the slope indicates that the line is angled downwards rather than upwards.

The point-slope form of a line is a useful way to write the equation of a line when you know one point on the line and the slope of the line. The formula is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a specific point on the line.
Let's break it down:

• \( m \) represents the slope of the line.
• \( x_1 \) and \( y_1 \) are the coordinates of a point on the line.

The formula derives from the definition of slope \((\frac{\Delta y}{\Delta x})\), ensuring our line fits through a certain point with the right slope. To find the specific equation of a line with negative slope passing through \((a, 0)\) and forming angle \(\theta\) with the x-axis, we replace \(m\) with \(-\tan(\theta)\) per our angle's calculation. This method is an easy way to write a line equation given any point and slope.

Trigonometric functions can define relationships in right-angled triangles, and they also help describe angles in relation to circles and, importantly, slopes of lines. Consider the angle \( \theta \) a line makes with the x-axis. The slope of the line can be connected to this angle using the tangent function, \( \tan(\theta) \), which represents the ratio between the opposite and adjacent sides in a right-angled triangle.
When determining the line's equation with respect to an acute angle \( \theta \) in this problem, the slope becomes vital. Our exercise involves a negative slope, so you use \( -\tan(\theta) \) to indicate this downward direction. The trigonometric function provides the basis for connecting an angle to a linear slope, bridging the gap between geometry and algebra in one elegant formula.