Begin by using the Law of Sines if two angles and any side are given, or if two sides and an angle opposite one of them are given. Begin by using the Law of Cosines if two sides and their included angle are given, or if the three sides are given.
We can use the Law of Sines and the Law of Cosines to find missing measurements in triangles that are not right triangles. Finding all of the missing measurements is called solving the triangle. To do this, we have to determine which of the two laws must be used first.
Let's determine which law to use first and then solve the triangle.
Determining the Law
In △ R S T we are told that S= 106^(∘), r= 20, and t= 16.
We have information about two sides and their included angle. Therefore, we must start solving the triangle by using the Law of Cosines.
Solving the Triangle
We will find the missing side length and the missing angle measures one at a time.
Finding the Side Length
Let's consider the Law of Cosines and adapt it to our triangle.
s^2= r^2+ t^2-2 r tcos S
We can substitute the given values into this equation and solve for s.
Let's add the newly obtained information to our diagram.
Since now we have three side lengths and one angle measure, we can use the Law of Sines.
sin R/r=sin S/s
Let's substitute r= 20, S= 106^(∘), and s= 28.9 into the above equation and solve for R.
Finally, we can use the Triangle Angle Sum Theorem to find the measure of T.
42^(∘) + 106^(∘) + T = 180^(∘)
⇕
T= 32^(∘)
We have completely solved the triangle.
