Simply put, the angle of inclination is the measure of the space between two lines on a graph. Because lines on a graph are often drawn at a diagonal, this space is usually triangular in shape. Because all triangles are measured by their angles, this space between two lines must often be represented by "angles" of inclination. When the slope of a line can not be measured in the conventional way, we can use the angle of inclination because the angle of inclination and the slope of the line are actually equal.
A slope is a ratio of change from the vertical to the horizontal of a line on a graph. This is usually represented by the letter m. The larger the slope of a line is, the steeper it is. If a slope is represented by a negative number, then the line is not moving in an upward motion on the graph, it is moving in a downward motion.
On a regular graph, the x- and y-axis bisect each other on the perpendicular and form four right angles. On a graph where the only lines are x and y, inclination will always be 90 degrees. This is because inclination is the measure of the positive section of the x-axis (the upper two quadrants of a graph) until it hits a line. In this case, as the only other line is the y-axis, inclination spans the whole upper right quadrant of the graph making inclination 90 degrees. Any line that is horizontal has an inclination of 0 and any line that is vertical has an inclination of 90. You should note that horizontal lines mirror the x-axis and vertical lines mirror the y-axis.
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The tangent function is used in trigonometry to determine the measure of an angle in a triangle. Tangent only measures the angle made by the two lines of a triangle that are not the hypotenuse. This function should not be confused with the other tangent in mathematics that also has to do with slopes. That tangent is the point at which a slope touches a curve of a another function. In terms of the angle of inclination of a slope, tangent is only used to measure the angle and is not used in any other way.
Angle of Inclination
The angle of inclination of a slope is the measure of inclination from the x-axis to a line, or slope, on a graph. As with the measure of inclination on graph, this is the measure of the angle made between a positive section of the x-axis moving counter-clockwise until it hits the slope of the line. If the slope of the line is positive, it moves through the upper right quadrant of the graph and the angle will be small. If the slope of the line is negative, it moves through the upper left quadrant and the angle will be large. The tangent function is used to measure this angle and treats the x-axis as one line of a triangle and the slope of the line as the other tangential line. The slope of a line and the tangent will always be equal to each other.