Which of the following is a formula
that can be used to find the area of a triangle? A) one-half 𝑎𝑏 cos 𝐶, B)
one-half 𝑎𝑏 sin 𝐶, C) one-third 𝑎𝑏 sin 𝐶, D) one-fourth 𝑎𝑏 cos 𝐶, or E)
one-fourth 𝑎𝑏 sin 𝐶.
If we sketch a triangle and label
it 𝐴, 𝐵, and 𝐶, the side length opposite vertex 𝐴 is usually labelled with a
lower case 𝑎. The side length opposite vertex 𝐵
is labelled with a lower case 𝑏. And we label lower case 𝑐 the side
length opposite vertex 𝐶. We have to remember that a triangle
is half of a rectangle. And so, it’s unlikely that options
C through E would be the answer.
We noticed that options A and B are
dealing with the angle at vertex 𝐶, that’s this angle, and the lengths 𝑎 and
𝑏. At this point, we recognize that we
have two sides and an included angle. And we know that the height of this
triangle will be equal to 𝑏 times the sin of 𝐶. To use trigonometry to solve for
the area of a triangle, we take one-half times 𝑎 times 𝑏 times sin of 𝐶, which is
option B here.