For a triangle 𝐴𝐵𝐶, 𝑎 is equal
to two centimeters, 𝑏 is equal to five centimeters, and the measure of angle 𝐵 is
40 degrees. Find all possible measures of angle
𝐴 to the nearest degree.
In this question, we are given the
lengths of two sides of our triangle together with a measure of one of the
angles. And we are asked to find all the
possible measures of a second angle 𝐴. If triangle 𝐴𝐵𝐶 exists from the
measures given, we know that we can find one possible measure of angle 𝐴 using the
law of sines. This states that sin 𝐴 over 𝑎 is
equal to sin 𝐵 over 𝑏, which is equal to sin 𝐶 over c, where uppercase 𝐴, 𝐵,
and 𝐶 are the measures of the three angles and lowercase 𝑎, 𝑏, and c are the
lengths of the sides opposite the corresponding angles.
Substituting the values we’re given
in this question into the law of sines gives us sin 𝐴 over two is equal to sin of
40 degrees over five. We can multiply through by two such
that sin 𝐴 is equal to two multiplied by sin 40 degrees all divided by five. Typing this into our calculator
gives us 0.2571 and so on. We can then take the inverse sine
of both sides of this equation, giving us 𝐴 is equal to 14.89 and so on. We are asked to give our answer to
the nearest degree. So one possible measure of angle 𝐴
is 15 degrees.
To work out whether there is a
second possible solution, we recall that the sin of 180 degrees minus 𝜃 is equal to
sin 𝜃. As 180 degrees minus 15 degrees is
165 degrees, then the sin of 165 degrees must equal the sin of 15 degrees. This means that we need to check
whether 165 degrees is a valid solution. Since angles in a triangle sum to
180 degrees and 165 plus 40 is equal to 205, 165 degrees is not a valid solution to
this problem. We can therefore conclude that the
only possible measure of angle 𝐴 is 15 degrees to the nearest degree.
It is worth noting that we could’ve
established that there was only one possible solution from the measurements given in
the question, since if angle 𝐵 is acute and side length 𝑏 is greater than side
length 𝑎, then only one triangle exists. And this means that there is only
one possible measure of angle 𝐴.