Solving Non-Right Triangles
A non-right, or oblique, triangle has no right angles.
Yet trigonometry—a subject whose rules are generally based on right
triangles—can still be used to solve a non-right triangle. You need
different tools, though. Enter the laws of sines and cosines.
In an oblique triangle, there are six unknowns: the three
angle measures and the three side lengths. To solve an oblique triangle
you need one of the following sets of information:
Two sides and an angle opposite one of the known sides
angles and any side
sides and their included angle
If you know either (1) or (2), you can use the law of
sines to solve the triangle. If you know (3) or (4), you must tag-team
with the law of cosines and then the law of sines to find the solution.
The Law of Sines
The law of sines is based on the proportionality of sides
and angles in triangles. The law states that for the angles of a
non-right triangle, each angle of the triangle has the same ratio of
angle measure to sine value.
If you are given the lengths of two sides and the measure
of an angle opposite one of those sides, you can use the law of
sines to find the other opposite angle. The measure of the third angle
can be easily found using the fact that the sum of the angles of
a triangle is 180º. Finally, you can use the law of sines again
to find the length of the unknown side. Here’s an example:
triangle ABC, a = 5, b =
6, and = 65Âº. Solve the triangle.
by plugging the
values of a
into the law of sines:
. You can do this
by using the rule that all interior angles of a triangle add up to
Last, find c
by plugging in a
into the law of sines:
The triangle is solved.
The Law of Cosines
The law of cosines offers a different way of solving non-right
triangles and can be used when you don’t have the information necessary
to use the law of sines. This is the law of cosines:
If you look carefully at the law of cosines, you should
see a resemblance to the Pythagorean theorem. In fact, for right
triangles, the law of cosines simplifies to the Pythagorean theorem.
Try it yourself. The last term drops out (since cos 90 = 0) and
you’re left with the familiar formula of c2 = a2 + b2.
If you’re curious, the 2ab cos(C)
term compensates for the lack of a right angle.
The law of cosines allows you to solve any triangle for
which you know any three of the four unknowns in the formula. There
are two ways you might know three of the four unknowns:
If you know two sides and their included angle, use
the law of cosines to find the length of the third side. Then use
the law of sines to complete the triangle.
you know the lengths of all three sides, use the law of cosines
to find the measure of one angle. Then use the law of sines to complete
We’ll just show one example problem, because using the
law of cosines is basically just plugging values into the formula
and solving it.
triangle ABC if a = 4, b =
7, and c = 10.
At this point, you can use the law of sines to find that A ≈
18.20º and B ≈ 33.12º.