Let's begin by drawing â–³ ABC and labeling the lengths of the sides. We will also color code the opposite angles and sides. It will help us use the Law of Sines and Law of Cosines later.
Let's find the length of the side a, and measures of ∠B, ∠C one at a time.
Finding a
We are given two sides and their included angle. Therefore, we can use the Law of Cosines to find the third side.
a^2 = b^2 + c^2 - 2bccos A
Let's substitute b= 5, c= 3, and A = 116^(∘).
Now that we know the length of the side a, we can find m ∠B using the Law of Sines.
sin A/a =sin B/b
Let's substitute a= 6.9, b= 5, and m ∠A = 116^(∘) to isolate sin B.
Finally, to find m ∠C we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to 180^(∘).
116^(∘) + 41^(∘) + m ∠C= 180^(∘) ⇔ m ∠C ≈ 23^(∘)
Completing the Triangle
With all of the angle measures, we can complete our diagram.
Extra
The Law of Cosines can be visualized by drawing a triangle and labeling angles and sides.
Similarly, the Law of Sines can also be visualized by drawing a triangle and labeling angles and sides.
