Step-by-step explanation:
We notice that if the gradient changes, then the value of \(\theta\) also changes, therefore the angle of inclination of a line is related to its gradient. We know that gradient is the ratio of a change in the \(y\)-direction to a change in the \(x\)-direction:
\[m=\frac{\Delta y}{\Delta x}\]
From trigonometry we know that the tangent function is defined as the ratio:
\[\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}\]
And from the diagram we see that
\begin{align*} \tan \theta &= \dfrac{\Delta y}{\Delta x} \\ \therefore m &= \tan \theta \qquad \text{ for } \text{0}\text{°} \leq \theta < \text{180}\text{°} \end{align*}
Therefore the gradient of a straight line is equal to the tangent of the angle formed between the line and the positive direction of the \(x\)-axis.
