For any △ABC, let the lengths of the sides opposite A, B, and C be a, b, and c, respectively.
The relates the of each angle to the length of the opposite side.
asinA=bsinB=csinC
Let's start by finding
m∠A. Then, we will use the above law to find the values of
a and
c. We will find them one at a time.
Finding A
Consider the that satisfies the given conditions.
From the we know that the sum of the angles in a triangle is equal to
180∘. With this information we can find
A.
m∠A+18∘+142∘=180∘⇕m∠A=20∘
Finding a
Let's mark the measure of the angle A on the graph.
We know that the length of a side is
20 and that the measure of its opposite angle is
18∘. We also know that the measure of the angle that is opposite to the side we want to find is
20∘. With this information and using the
Law of Sines, we can write an equation in terms of
a.
20sin18∘=asin20∘
Let's solve the above equation for
a using the .
20sin18∘=asin20∘
sin18∘⋅a=sin20∘⋅20
a=sin18∘sin20∘⋅20
a=22.136008…
a≈22.1
Finding c
Consider the triangle with the new information.
We know that the length of a side is
20 and that the measure of its opposite angle is
18∘. We want to find the length of the side that is opposite to the angle whose measure is
142∘. We can use the
Law of Sines again!
20sin18∘=csin142∘
Let's solve the above equation for
c using the
Cross Product Property.
20sin18∘=csin142∘
sin18∘⋅c=sin142∘⋅20
c=sin18∘sin142∘⋅20
c=39.846447…
c≈39.8