Question

if $\theta$, $0 < \theta < pi$, is the angle between a horizontal ray directed to the right (say, the positive x-axis) and a nonhorizontal, nonvertical line l, show that the slope m of l equals $\tan\theta$, then the angle $\theta$ is called the inclination of l. hint: see the illustration, where line m has been drawn parallel to l and passing through the origin. use the fact that m intersects the unit circle at the point $(cos\theta, sin\theta)$.
equate the slopes of two parallel lines.
= the slope of line m
$=\square$
find the slope of line m using the two points.

Explanation:

Step1: Calculate slope of line M

Slope formula: $m_M = \frac{y_2 - y_1}{x_2 - x_1}$
Points: $(0,0)$ and $(\cos\theta, \sin\theta)$
$m_M = \frac{\sin\theta - 0}{\cos\theta - 0} = \frac{\sin\theta}{\cos\theta}$

Step2: Simplify the slope

Trigonometric identity: $\frac{\sin\theta}{\cos\theta} = \tan\theta$

Step3: Equate slopes of parallel lines

Parallel lines have equal slopes: $m_L = m_M$

Answer:

The slope $m$ of line L is $\tan\theta$, so $m = \tan\theta$