Video Transcript
𝐴𝐵𝐶 is an obtuse triangle at 𝐴,
where 𝑏 equals 15 centimeters, tan of 𝐶 equals six over five, and the measure of
angle 𝐵 equals 27 degrees. Find the lengths 𝑎 and 𝑐, giving
the answer to the nearest integer.
It’s always a good idea to begin by
sketching a diagram. This doesn’t need to be to scale,
but it should be roughly in proportion. So we can check the suitability of
any answers we get. We’ll label the triangle such that
the obtuse angle is angle 𝐴. We’ll then add the other
information we know. Since we’ve been told that the tan
of 𝐶 equals six-fifths, we can solve this equation to find the angle at 𝐶. The inverse tan of the tan of 𝐶
equals 𝐶. And the inverse tan of six-fifths
is equal to 50.194 continuing degrees. Since we know we’re going to be
rounding in a later step, it’s best not to round at this point. This will prevent errors by
rounding too early.
Next, we can solve for the measure
of the angle at 𝐴. We know that the angles in a
triangle sum to 180. Therefore, the measure of angle 𝐴
will be equal to 180 degrees minus 27 degrees plus 50.194 continuing. Solving gives us 102.805 continuing
degrees. This angle is an obtuse angle; it
is greater than 90 degrees. Now, we have a non-right
triangle. And we know the lengths of one side
and all the angles. This means we can use the law of
sines to find any missing measurement. We don’t use the law of cosines
here as that requires a minimum of two known sides.
The formula for the law of sines is
𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵, which equals 𝑐 over sin 𝐶. Alternatively, this can be written
as sin 𝐴 over 𝑎 equals sin 𝐵 over 𝑏, which equals sin 𝐶 over 𝑐. Since we’re trying to solve for
side length, we’ll use the first form. Either form will work, but by using
the first form, we reduce the amount of rearranging we’ll need to do to solve. Before we move on, let’s label the
sides of our triangle. The side opposite angle 𝐴 will be
labeled with a lowercase 𝑎. The side opposite angle 𝐵 will be
lowercase 𝑏. And the side opposite angle 𝐶 will
be lowercase 𝑐.
First, let’s calculate the length
of side 𝑎. This means we’ll need 𝑎 over sin
𝐴 equals 𝑏 over sin 𝐵. We’re using 𝑏 over sin 𝐵 as we
know both value 𝑏 and the angle at 𝐵. Substituting what we know, we get
𝑎 over sin of 102.805 continuing is equal to 15 over sin of 27 degrees. Once again, we’ll use the exact
value for the measure at angle 𝐴. To solve, we’ll multiply both sides
of the equation by sin of 102.8 continuing. 𝑎 is therefore equal to 15 over
sin of 27 degrees times sin of 102.8 continuing degrees. Popping that into the calculator
gives us 𝑎 equals 32.2185 continuing. That value to the nearest integer
will be 32 centimeters.
We’ll follow the same process to
solve for the length of side 𝑐. Using 𝑐 over sin 𝐶 is equal to 𝑏
over sin 𝐵. Substituting what we know, we’ll
have 𝑐 over sin of 50.194 continuing degrees is equal to 15 over sin of 27
degrees. Then we’ll multiply through by sin
of 50.194 continuing degrees. 𝑐 equals 15 over sin of 27 degrees
times sin of 50.194 continuing degrees, which gives us 25.382 continuing. And to the nearest integer, side 𝑐
is then 25 centimeters.
