The angle between two lines and applications of trigonometry

Summary

This lesson uses trigonometry to measure the angle formed by two lines in the plane. It defines the inclination of a line and proves that its tangent is the slope; with the formula for the tangent of a difference it obtains the tangent of the angle between two lines from their slopes, which contains the perpendicularity condition as a special case, and with the dot product it gives a second expression, also valid for vertical lines, whose agreement with the first is Lagrange’s identity. It then characterizes the bisectors of two lines as the locus of the points equidistant from both, proves that they are perpendicular and uses them to obtain the incenter and the inscribed circle of a triangle. The last section gathers applications: the height of an inaccessible point by means of the law of sines, simple harmonic motion as the shadow of uniform circular motion, the range of a projectile, which is greatest at $45^\circ$ and the same for complementary angles, and the law of reflection as a property of the shortest path.

Learning objectives

Compute the inclination of a line and the angle between two lines, from their slopes or from their direction vectors, and recognize parallelism and perpendicularity as the extreme cases.
Obtain the equations of the bisectors of two lines as a locus of equidistant points, choose the internal bisector of a triangle and compute its incenter and its inscribed circle.
Solve problems of heights and distances with angles of elevation and of depression, combining the law of sines with the sum and difference formulas.
Describe simple harmonic motion, the range of a projectile and the reflection of light with trigonometric functions, and derive their properties from the identities of this unit.

The inclination of a line

In the Course I lesson on the equation of a line, the direction of a non-vertical line was measured by a number, its slope, and it was proved that two non-vertical lines are perpendicular if and only if the product of their slopes is $-1$ . That condition recognizes one particular angle, the right angle, without measuring any angle. In the lesson on trigonometric ratios it was also seen that the tangent of an angle is the slope of the line containing its terminal side. If this is so, the slope is the tangent of an angle, and the angles between lines should be computable with trigonometry.

The angle of inclination of a line is the angle $\alpha$ , with $0 \le \alpha < \pi$ , that the positive half of the horizontal axis forms with the line, measured counterclockwise from the axis to the line. Horizontal lines have inclination $0$ , and vertical lines, inclination $\frac{\pi}{2}$ .

More precisely, the parallel to the line through the origin contains the terminal side of some angle $\alpha$ , and also that of $\alpha + \pi$ , because the points $P(\alpha)$ and $P(\alpha + \pi)$ of the unit circle are opposite; exactly one of the angles of the form $\alpha + k\pi$ lies in $\ico{0}{\pi}$ , and that one is the inclination. Two lines are parallel or coincide if and only if they have the same inclination, because then, and only then, they have the same parallel through the origin.

LemmaInclination and slope

A line of inclination $\alpha$ is vertical if and only if $\alpha = \frac{\pi}{2}$ ; if it is not vertical, its slope is the tangent of its inclination. Conversely, the inclination of a line of slope $m$ is $\arctan m$ if $m \ge 0$ , and $\pi + \arctan m$ if $m < 0$ .

m = \tan \alpha

Everything is read on the parallel to the line through the origin, which contains the point $P(\alpha)$ of the unit circle. The figure uses the lines $y = \sqrt{3} x + 1$ and $y = -x$ .

Proof

$\dato{p}{O = \left( 0, 0 \right) \in r_0, \quad P(\alpha) = \left( \cos \alpha, \sin \alpha \right) \in r_0}$
Let $r$ be a line of inclination $\alpha$ , and let $r_0$ be its parallel through the origin. By the definition of the inclination, $r_0$ contains the terminal side of $\alpha$ , and in particular the point $P(\alpha)$ ; if $r$ is not vertical, $r_0$ has the same slope, because two parallel lines have the same slope, according to the Course I lesson on the equation of a line. In the figure, $\alpha = \frac{\pi}{3}$ .
$0 \leq \alpha < \pi, \quad \resaltar{\cos \alpha = 0} \leftrightarrow \alpha = \frac{\pi}{2}$
The line $r_0$ is vertical if and only if its two points have the same abscissa, that is, if $\cos \alpha = 0$ . In $\ico{0}{\pi}$ , the only zero of the cosine is $\frac{\pi}{2}$ ; and $r$ is vertical exactly when its parallel is.
$m = \frac{\sin \alpha - 0}{\cos \alpha - 0} = \dato{m}{\resaltar{\tan \alpha}}$
If $r$ is not vertical, its slope is that of $r_0$ , which is the quotient of the difference of the ordinates by that of the abscissas of its two points.
$\alpha = \begin{cases} \resaltar{\arctan m} & m \geq 0 \\ \resaltar{\pi + \arctan m} & m < 0 \end{cases}$
Conversely, let $m$ be the slope of $r$ . By the step above, the inclination $\alpha$ is an angle in $\ico{0}{\pi}$ whose tangent is $m$ . According to the lesson on trigonometric equations, the angles with tangent $m$ are the numbers $\arctan m + k\pi$ , and exactly one of them lies in $\ico{0}{\pi}$ : since $\arctan m$ lies in $\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}}$ , that angle is $\arctan m$ if $m \ge 0$ , and $\pi + \arctan m$ if $m < 0$ . In the figure, the line $y = -x$ has inclination $\pi - \frac{\pi}{4} = \frac{3\pi}{4}$ .

For example, the line $y = \sqrt{3} x + 1$ has inclination $\arctan \sqrt{3} = \frac{\pi}{3}$ , and the line $y = -x$ , inclination $\pi + \arctan \left( -1 \right) = \frac{3\pi}{4}$ . The sign of the slope thus distinguishes the lines that rise, with acute inclination, from those that fall, with obtuse inclination.

In the rest of this lesson

The angle between two lines
The angle via direction vectors
The bisectors of two lines
The incenter of a triangle
Applications of trigonometry
Heights and distances
Simple harmonic motion
The range of a projectile
The law of reflection
Solved problems
The angle between two lines, in two ways
The lines that form 45 degrees with a given line
The incenter of a triangle given by its sides
The meeting of two oscillations
The angle to hit a target
Proposed problems
Three arctangents that add up to pi
The height of a tower seen from a building
The position of a boat from the angle between two lighthouses
The bisectors and the half-sum of the inclinations
The reflector of two perpendicular mirrors