ANGLE BETWEEN TWO LINES
When
two lines intersect, the angle between them is defined as the angle through
which one of the lines must be rotated to make it coincide with the other line.
For example, the angle
(the Greek letter phi) in figure 17 is the
acute angle between lines L, and L_{2}.
Referring
to figure 17,
We
will determine the value of + directly from the slopes of lines L, and L_{2},
as follows:
Figure 17.Angle between two lines.
We
obtain this result by using the trigonometric identity for the tangent of the
difference between two angles. Trigonometric identities are discussed in
chapter 6 of Mathematics, Volume 2A,
NAVEDTRA 10062.
Recalling that the tangent of the angle of inclination is
the slope of the line, we have
such that
Substituting these expressions in the tangent formula
derived in the above discussion, we have
EXAMPLE: Referring to figure 17, find the acute angle
between the two lines that have m, =
and m_{2} = 2 for their
slopes.
SOLUTION:
such that
or, referring to Appendix 1,
NOTE: To find the obtuse angle between lines L, and L_{2}, just
subtract the acute angle between L, and L_{2} from 180
°. Referring to figure 17,
If the obtuse angle in the previous example was to be
found, then
If one of the lines was parallel to the Y axis, its slope
would be infinite. This would render the slope formula for tan + useless,
because an infinite value in both the numerator and denominator
of the fraction
produces an indeterminate form. 1 + m,m2
However, if only one of the lines is known to be parallel
to the Y axis, the tangent of + may be expressed by another method. Suppose
that L_{2} (fig
17) was parallel to the Y axis. Then we would have
If L, has a positive slope, then the
acute angle between L, and L_{Z} would
be found by
If L, has a negative slope, then
As before, the obtuse angle can be found by using
PRACTICE PROBLEMS:
1. Find the acute angle between the two lines that have m,
= 3 and m_{2} = 7 for their slopes.
2. Find the acute angle between two lines whose slopes are
m, = 0 and m_{2} = 1. (m, = 0 signifies that line L, is horizontal and the formula still holds.)
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3. Find the acute angle between the Y axis and a line with
a slope of m =  8.
4. Find the obtuse angle between the X axis and a line
with a slope of m =  8.
ANSWERS:
