Solution of Triangles
In a triangle, there are six elements or parts, three sides and three
angles. From plane geometry, we know that if three of the elements are given,
at least one of which must be a side, then the other three elements can be
uniquely determined. If only three angles are given, then no unique solution of
the triangle is possible but unlimited number of equiangular triangles can be
found, the three angles of which are equal to the three given angles. The
method of determining the unknown parts from the known parts, is called solving
a triangle.
Again, it is evident that the three given parts
cannot be independent or arbitrary. For instance, in a triangle, two of its
angles can not be obtuse or the sum of any two sides cannot be less than or
equal to the third.
A triangle is called a right-angled triangle or
right triangle if one of its angle is a right angle otherwise, the triangle is
called an oblique triangle. We start with the solution of right-angled
triangles and the solution of oblique triangle will follow.
Whenever found convenient, logarithms should be
used.
Solution of right-angled triangle
In this case, one angle is equal to 90 degree. If
any two (at least one of which is a side) of the remaining five parts are given
then the other unknown parts of the triangle can be determined.
If ABC be a right-angled triangle, where C = 90
degree then for solving the triangle, we have the following relations:
C = 90 degree, A B = 90 degree,
c^2 = a^2 b^2.
sin A =a/c, sin B = b/c, tan A =a/b = cot B.
Solution
of oblique triangles
The various cases
that may arise in solving oblique triangles are the following:
a) Three angles are given:
In this case, no definite values of the sides can
be found but the ratio of the sides is determined.
b) Three sides are given
In this case, unique solution of the triangle is
possible.
c) Two sides and the included angle between them
are given
In this case, triangle is unique.
d) Two angles and one side are given
The triangle is also unique in this case.
e) Two sides and the angle opposite to one of them
are given
In this case, no triangle or more than one
triangle is possible (ambiguous case).
General Remarks
Different trigonometric formulae may be used in
different cases. The same problem may also be solved by using different
formulae and by different methods. If no additional data are given in the
problem, the formula, which is most suitable, should be chosen. If additional
data are supplied, then the formula, which will fit in with the given data, must
be selected. It has been found that for logarithmic computations tangent
formula is most suitable.
The difference in the value of tan x is larger
than that of tabular sin x or cos x. Hence, a small error in the tabular
tangent will not affect the value of x so much as in the case tabular sine or
cosine. In fact, 4-figure logarithmic tangent table gives more accurate result
than even the 7-figure logarithmic sine or cosine table.
Note: If the angle is determined by the sine
formula, then ambiguity may arise in the result. For example, if sin A = 1/2,
then A = 30 degree or 150 degree, both are equally possible for being an angle
of a triangle. This difficulty will not arise, when the angle is determined by
cosine or tangent formula.
