OBLIQUE ANGLE
TRIGONOMETRY
Thus far our study of trigonometry has been limited to right
triangles. Application of trigonometry to
oblique triangles
is studied in this chapter.
Oblique triangles are often encountered in shop problems.
Distances between holes, lengths of sides, and angles between centerlines are
usually indicated in the layout of machine parts. The oblique triangles formed
by sides and centerlines may be solved by construction of right triangles on
the obliques or by the use of special formulas given in this chapter.
Oblique triangles are triangles in which none of the angles are
right angles.
Acute
oblique triangles
have
all angles less than 90 degrees.
Obtuse oblique triangles s
ave one angle greater than 90 degrees.
One way to solve a problem involving an oblique triangle is by
reconfiguring the problem into one involving right triangles. Auxiliary lines
are added to construct the necessary right triangles by dropping perpendiculars
from angle vertices to their opposite sides. This construction is shown in
Figure 21.1 for an acute oblique triangle and in Figure 21.2 for an obtuse
oblique triangle. The resulting right triangles may then be solved by applying
the rules for right angle trigonometry.
In the case of acute oblique triangles typified in Figure 21.1,
the altitudes all fall
within triangle
ABC
and
six right triangles are formed:
ADC
and
ADB
;
BFA
and
BFC
;
CEA
and
CEB.
For obtuse oblique triangles such
as shown in Figure 21.2,
two of the altitudes,
C
′
D
′
and
B
′
F
′
, fall outside triangle
A
′
B
′
C
′
, and one altitude,
A
′
E
′
, falls within the triangle.
Here, too, six right triangles are formed;
A
′
F
′
B
′
and
A
′
D
′
C
′
,
B
′
D
′
C
′
and
B
′
E
′
A
′
,
C
′
E
′
A
′
and
C
′
F
′
B.
′