SAT Subject Test Mathematics Level 2

LAW OF SINES AND LAW OF COSINES

If  you know    any two angles  and one side    of  a   triangle,   you can figure  out the other   sides   by

using   the Law of  Sines.  For any triangle,   the side    lengths are proportional    to  the sines   of  the

opposite    angles:

If  you know    two sides   of  a   triangle    and the angle   between them,   you can figure  out the third

side    by  using   the Law of  Cosines,    which   is  a   more    general version of  the Pythagorean

theorem:

c^2     =   a^2     +   b^2     −   2ab cos C

You might not think that this is a trigonometry question at first. There’s no “sin,” “cos,” or any other
explicit trig function mentioned in the stem. And the triangle’s not a right triangle. But trig is the
tool you have to use to answer this question. This is a case of “solving a triangle”; that is, finding the
length of one or more sides.

With the Law of Sines and the Law of Cosines and a calculator, you can solve almost any triangle. If
you know any two angles—which of course means that you know all three—and one side, you can
use the Law of Sines to find the other two sides. If you know two sides and the angle between them
—remember, it must be the angle between them—you can use the Law of Cosines to find the third
side.

In Example 6, what you’re given is two angles and a side, so you’ll use the Law of Sines, which says
simply that the sines are proportional to the opposite sides. Here the side you’re looking for, BC, is

(D) 8.5

(E) 8.9