Derivation of the Law of Sines: To calculate side or angle lengths of right triangles, you can set up a trigonometric ratio using sine, cosine, or tangent. However, if the triangle does not include a right angle, these basic trigonometric ratios do not apply. Triangles that do not have a right angle are called oblique triangles. Although the basic trig ratios do not apply, they can be modified to cover oblique triangles.
In this lesson, we will investigate how to apply the sine function to an oblique triangle. Consider the following triangle,  ABC. Remember that the side and angle of a triangle that share the same name are always across from each other.
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Using the smaller triangle on the left that includes angle A and sides b and h,
we can set up an equation involving sine. |
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Using the triangle on the right half that includes angle B and sides a and h,
we can set up and equation involving sine. |
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Both of these equations involve “h”.
Solve both equations for “h”. |
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Set the two expressions for “h” equal to each other. |
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Divide both sides by ab. |
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Reduce each fraction. |
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Final equation that uses the sine function for oblique triangles. |
It can be shown in a similar example that this also applies to side c and angle C. This results in the Law of Sines for oblique triangles which is summarized in the box below.
One of the benefits of the Law of Sines is that not only does it apply to oblique triangles, but also to right triangles. Let’s use a familiar right triangle: the 30°, 60°, 90° triangle shown below:
 To “ solve the triangle” means to find all angle and side lengths. You must have enough information to define a unique triangle. This will take us back to investigating what information was needed to prove triangle congruencies in Geometry.
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1.
AAS
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2.
SSS
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3.
SAS
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2 angles & non-included side |
all three sides |
two sides & included angle |
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To use the Law of Sines, one needs to be able to set up ONE complete ratio, that is, one needs an angle and the length of its opposing side as givens. If this information is not given, the Law of Sines cannot be used. Examination of the above five cases shows that the Law of Sines CAN be used with: AAS, ASA (because if two angles of a triangle are known, then the third can be found by subtracting from 180°), and SSA (with a little additional work).
Recall that SSA this was NOT enough information to prove that a unique triangle existed. Further work is needed to apply the Law of Sines in this case. This is referred to as the Ambiguous Case and will be addressed in Part II of this lesson
The steps used to calculate angle and side lengths of an oblique triangle are outlined below. Before Law of Sines can be used, one has to check and see if it even applies to the given triangle information.
- Draw a picture of the triangle and label all knowns.
- Do I have AAS or ASA? If yes, you can set up the Law of Sines and solve for the missing angle or side.
A quick rough check of your answers can come from a well-known geometric theorem that states:
There is one case where you will have a problem if you use the Law of Sines. Since the sine function is positive in both the first and second quadrants, the Law of Sines will never give an obtuse angle as an answer. Do not find the largest angle with the Law of Sines, instead, use the Law of Cosines.
Refering back to at the beginning of this lesson where the base was side c, we discovered two different expressions for the height.
By tilting the original triangle so that side b is used as a base, the height would equal (a sin C):
In each of these area equations, each of the variables is used; one as an angle measure, the other two as side lengths. Any of these forms is a correct equation that can be used to calculate the area of an oblique triangle.
Let's Practice solving some triangles!
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