Solving Non-Right Triangles
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:
  1. Two sides and an angle opposite one of the known sides
  2. Two angles and any side
  3. Two sides and their included angle
  4. All three sides
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:
In triangle ABC, a = 5, b = 6, and = 65º. Solve the triangle.
First, find by plugging the values of a, b, and into the law of sines:
Next, find . You can do this by using the rule that all interior angles of a triangle add up to 180º:
Last, find c by plugging in a, b, and 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:
  1. 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.
  2. If 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 the triangle.
We’ll just show one example problem, because using the law of cosines is basically just plugging values into the formula and solving it.
Solve triangle ABC if a = 4, b = 7, and c = 10.
First, find .
At this point, you can use the law of sines to find that A ≈ 18.20º and B ≈ 33.12º.
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