𝐴𝐵𝐶 is a triangle where 𝑎 is 96
and the measure of the angle at 𝐵 is equal to three times the measure of the angle
at 𝐴, which is equal to 90 degrees. Find length 𝑐 giving the answer in
terms of sine.
It’s always sensible to begin by
sketching a diagram out. It 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’re told that the measure of the
angle at 𝐵 is equal to three times the measure of the angle at 𝐴 and that these
two expressions are equal to 90 degrees. This of course means that the
measure of the angle at 𝐵 is 90 degrees. It’s a right angle.
Let’s look at 𝐴 then. Since three times the measure of
the angle 𝐴 is 90 degrees, we can work out the measure of the angle at 𝐴 by
dividing through by three. And doing so, we can see that the
measure of the angle at 𝐴 is 30 degrees. We can also work out the measure of
the angle at 𝐶. Angles in a triangle sum to 180
degrees. So we can subtract 90 and 30 from
180. And that tells us the measure of
the angle at 𝐶 is 60 degrees.
Now, let’s label the sides of the
triangle. The side opposite angle 𝐴 is
lowercase 𝑎. The side opposite angle 𝐵 is
lowercase 𝑏. And the side opposite angle 𝐶 is
lowercase 𝑐. We want to calculate the length of
𝑐. Now normally, we could use right
angle trigonometry here. However, the question has asked us
to calculate it in terms of sine.
Instead, we’ll use the law of sines
that’s 𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵, which equals 𝑐 over sin 𝐶. Or alternatively, sin 𝐴 over 𝑎
equals sin 𝐵 over 𝑏, which equals sin 𝐶 over 𝑐. Since we’re trying to calculate the
length of a side, we’ll use the first form. In fact, we can actually use either
form of this formula. However, we’ll need to do less
rearranging if we use the first one this time.
Remember, we usually only need to
use two parts of this formula. Here, we know the length of the
side 𝑎 and we’re trying to find an expression for the side marked 𝑐. We’re going to use 𝑎 over sin 𝐴
and 𝑐 over sin 𝐶 then.
Let’s substitute the values from
our triangle into the formula. We get 96 over sin 30 is equal to
𝑐 over sin 60. We want to make 𝑐 the subject. So we’ll multiply both sides of
this equation by sin 60. And that gives us 𝑐 is equal to 96
over sin 30, all multiplied by sin 60. Remember, we can write sin 60 as
sin 60 over one. And then, when we multiply the
numerator of the first fraction by the numerator of the second fraction, we get 96
sin 60. And multiplying the two
denominators, we get sin 30.
We were told to leave our answer in
terms of sine. So we’ve finished. 𝑐 is equal to 96 sin 60 over sin