What is the size of angle 𝐴, in
degrees, in the triangle shown?
In this triangle, we see the angle
𝐴 marked out here, as well as the interior angle of 54 degrees. Opposite this 54-degree angle is a
side length of 8.4 centimeters. And then opposite angle 𝐴 is a
side length of 9.6 centimeters. We see then that this triangle is
well set up for us to use the sine rule to solve for this angle 𝐴. The sine rule applies to any
triangle. It says that if the interior angles
and the corresponding side lengths are marked out as shown, then the sine of any of
those interior angles divided by the corresponding side length is equal to that same
ratio for any of the other pairs of angles and sides.
So as we think about applying the
sine rule to our triangle over here, specifically to solve for angle 𝐴, we remind
ourselves that angle 𝐴 corresponds to the side length of 9.6 centimeters and the
angle of 54 degrees corresponds to 8.4 centimeters. So then the ratio of the sin of the
unknown angle 𝐴 to 9.6 centimeters is equal, by the sine rule, to the sin of 54
degrees divided by 8.4 centimeters.
And now what we want to do is to
isolate this angle 𝐴. To do this, we’ll first multiply
both sides of the equation by 9.6 centimeters, canceling that factor on the left and
then giving us this expression here. Notice that on the right-hand side
of our equation, these units of centimeters cancel from numerator and
And now what we need to do is to
invert or undo the application of the sine function on the angle 𝐴 we want to solve
for. We do this by applying what’s
called the arc sine or the inverse sine to the sin of 𝐴. And then to maintain our equality,
we apply the same inverse sine function to the right-hand side of our
expression. When we take the inverse sine of
the sin of the angle 𝐴, what remains is simply the angle 𝐴.
Our final step is to evaluate the
right-hand side of this expression. And just as the sine function is
standard on any scientific calculator, so the inverse sine function will be as
well. When we evaluate this expression
and find our answer in degrees, to two significant figures, it’s 68 degrees. That’s the size of the angle 𝐴 in
the triangle shown.