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METHODS OF SOLVING TRIANGLES | Up
Engineering Aid 3 - Beginning Structural engineering guide book | Next
Solution by Laws of Cosines |
determine the length of side b. You could do this
as previously described by applying
However, the fact that side b is larger than
side a means that tan B is larger than 1 (you recall
that any angle larger than 45° has a tangent larger
than 1).
You know that the cotangent is the reciprocal
function of the tangent. Therefore, if
it follows that
A table of natural functions tells you that cot
53°08' = 0.74991. Therefore,
Acute Angle of Right
Triangle by Sine or Cosine
If you know the length of the hypotenuse and
length of a side of a right triangle, you can
determine the size of one of the acute angles by
applying the sine or the cosine of the angle.
Suppose that for the triangle shown in figure 1-23,
you know that the hypotenuse, c, is 5.00 ft long
and that the length of side a is 3.00 ft long. You
want to determine the size of angle A. Side a is
opposite angle A; therefore,
A table of natural functions tells you that an angle
with sine 0.6 measures (to the nearest minute)
36°52'.
Suppose that, instead of knowing the length
of a, you know the length of b (4.00 ft). Side b
is the side adjacent to angle A. You know that
A table of natural functions tells you that an angle
with cosine 0.8 measures 36°52'.
If you know the size of one of the acute
angles in a right triangle and the length of
the side opposite, you can determine the
length of the hypotenuse from the sine of the
angle. Suppose that for the triangle shown in
figure 1-23, you know that angle A = 36°52' and
side a = 3.00 ft.
If you know the size of one of the acute
angles in a right triangle and the length of the side
adjacent, you can determine the length of the
hypotenuse from the cosine of the angle. Suppose
that for the triangle in figure 1-23, you know that
angle A = 36°52' and side b = 4.00 ft.
Tables show that cos 36°52' = 0.80003. There-
fore,
Solution by Law of Sines
For any triangle (right or oblique), when you
know the lengths of two sides and the size of the
angle opposite one of them, or the sizes of two
angles and the length of the side opposite one of
them, you can solve the triangle by applying the
law of sines. The law of sines (which is explained
and proved in NAVPERS 10071-B, chapter 5)
states that the lengths of the sides of any triangle
are proportional to the sines of their opposite
angles. It is expressed in formula form as follows:
In the triangle shown in figure 1-24,
LA = 41°24', a = 8.00 ft, and b = 12.00 ft. If
it follows that
Figure 1-24.—Oblique triangle (law of sines).
1-21
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