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.
Law: Sines Missing Parts: N = 53^(∘), m ≈ 28.4, p ≈ 38.2
Practice makes perfect
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 △ M N P we are told that M= 47^(∘), P= 80^(∘), and n= 31.
We have information about two angles and one side. Therefore, we must start solving the triangle by using the Law of Sines.
Solving the Triangle
We will find the missing angle measure and the missing side lengths one at a time.
Finding the Angle Measure
We can use the Triangle Angle Sum Theorem to find the measure of N.
47^(∘) + N + 80^(∘) = 180^(∘)
⇕
N= 53^(∘)
Finding the Side Lengths
Let's add the newly obtained information to our diagram.
Since now we have three angle measures and one side length, we can use the Law of Sines.
sin M/m=sin N/n
Let's substitute n= 31, N= 53^(∘), and M= 47^(∘) into the above equation and solve for m.
Let's use the Law of Sines once again and find p.
sin N/n = sin P/p
Let's substitute n= 31, N= 53^(∘), and P= 80^(∘) into the above equation and solve for p.
