METHODS OF SOLVING TRIANGLES
To “solve” a triangle means to determine one
or more unknown values (such as the length of
a side or the size of an angle) from given known
values. Here are some of the methods used.
Acute Angle of Right
Triangle by Tangent
Pythagorean Theorem
When you know the lengths of two sides of
a right triangle, or its hypotenuse and one side,
you can determine the length of the remaining
side, or the length of the hypotenuse, by applying
the Pythagorean theorem. The Pythagorean
theorem states that the square of the length of the
hypotenuse of any right triangle equals the sum
of the squares of the lengths of the other two
sides.
Figure 1-23 shows a right triangle with acute
angles A and B and right angle C. Sides opposite
A and B are designated as a and b; the hypotenuse
(opposite C) is designated as c. Side a measures
3.00 ft, side b measures 4.00 ft, and the hypot-
enuse measures 5.00 ft. Any triangle with sides
and hypotenuse in the ratio of 3:4:5 is a right
triangle.
If C2 = az + b2, it follows that c = i=.
The formulas for solving for either side, given the
other side and the hypotenuse; or for the
hypotenuse, given the two sides, are
Figure 1-23.-A right triangle.
One of the angles in a right triangle always
measures 900. Because the sum of the three angles
in any triangle is always 180°, it follows that each
of the other two angles in a right triangle must
be an acute (less than 90°) angle. Also, if you
know the size of one of the acute angles, you can
determine the size of the other from the formulas
A = (90° – B) and B = (90° – A).
In any right triangle in which you know the
lengths of the sides, you can determine the size
of either of the acute angles by applying the
tangent of the angle. Take angle A in figure 1-23,
for example. You know that
Reference to a table of natural tangents shows that
an angle with tangent 0.75 measures to the nearest
minute, 36°52'.
Side of Right Triangle by Tangent
If you know the length of one of the sides of
a right triangle and the size of one of the acute
angles, you can determine the length of the other
side by applying the tangent. Suppose that for the
triangle shown in figure 1-23 you know that angle
A measures 36°52' and that side b measures 4.00
ft. You want to determine the length of side a.
Since
Side of Right Triangle by Cotangent
Suppose that for the triangle shown in figure
1-23, you know that angle B measures 53°08´ and
that side a measures 3.00 ft. You want to
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