College Trigonometry, 2011a

912 Applications of Trigonometry
enough to determine if the angle in question is acute or obtuse. Since both authors of the
textbook prefer the Law of Cosines, we proceed with this method first. When using the Law
of Cosines, it’s always best to find the measure of the largest unknown angle first, since this
will give us the obtuse angle of the triangle if there is one. Since the largest angle is opposite
the longest side, we choose to find α first. To that end, we use the formula cos(α) = b2 +c 2 −a 2
2bc
and substitute a =7,b = √ 53 − 28 cos (50 ◦ )andc =2. Weget 3
cos(α) = 2 − 7 cos (50◦ )
√
53 − 28 cos (50 ◦ )
Since α is an angle in a triangle, we know the radian measure of α must lie between 0 and π
radians. This matches the range of the arccosine function, so we have
(
)
2 − 7 cos (50 ◦ )
α = arccos √ radians ≈ 114.99 ◦
53 − 28 cos (50 ◦ )
At this point, we could find γ using γ = 180 ◦ − α − β ≈ 180 ◦ − 114.99 ◦ − 50 ◦ =15.01 ◦ ,
that is if we trust our approximation for α. To minimize propagation of error, however, we
could use the Law of Cosines again, 4 in this case using cos(γ) = a2 +b 2 −c 2
2ab
. Plugging in a =7,
b = √ (
)
53 − 28 cos (50 ◦ 7−2cos(50
)andc =2,wegetγ = arccos
◦ )
√ radians ≈ 15.01 ◦ .We
53−28 cos(50 ◦ )
sketch the triangle below.
β =50 ◦
a =7
c =2 α ≈ 114.99 ◦ γ ≈ 15.01 ◦
b ≈ 5.92
As we mentioned earlier, once we’ve determined b it is possible to use the Law of Sines to
find the remaining angles. Here, however, we must proceed with caution as we are in the
ambiguous (ASS) case. It is advisable to first find the smallest of the unknown angles, since
we are guaranteed it will be acute. 5 In this case, we would find γ since the side opposite γ
is smaller than the side opposite the other unknown angle, α. Using the angle-side opposite
pair (β,b), we get sin(γ) sin(50
2
=
◦ )
√ . The usual calculations produces γ ≈ 53−28 cos(50 ◦ ) 15.01◦ and
α = 180 ◦ − β − γ ≈ 180 ◦ − 50 ◦ − 15.01 ◦ = 114.99 ◦ .
2. Since all three sides and no angles are given, we are forced to use the Law of Cosines. Following
our discussion in the previous problem, we find β first, since it is opposite the longest side,
b. We get cos(β) = a2 +c 2 −b 2
2ac
= − 1 5 ,sowegetβ = arccos ( − 1 5)
radians ≈ 101.54 ◦ . As in
3 after simplifying . . .
4 Your instructor will let you know which procedure to use. It all boils down to how much you trust your calculator.
5 There can only be one obtuse angle in the triangle, and if there is one, it must be the largest.