8.10: Laws of Sines and Cosines
What if you wanted to solve for a missing side in a nonright triangle? How could you use trigonometry to help you? After completing this Concept, you'll be able to use the Law of Sines and the Law of Cosines to answer questions like these.
Watch This
CK12 Foundation: Chapter8LawsofSinesandCosinesA
James Sousa: The Law of Cosines
Guidance
Law of Sines: If
Looking at a triangle, the lengths
Use Law of Sines when given:
 An angle and its opposite side.
 Any two angles and one side.
 Two sides and the nonincluded angle.
Law of Cosines: If
Even though there are three formulas, they are all very similar. First, notice that whatever angle is in the cosine, the opposite side is on the other side of the equal sign.
Use Law of Cosines when given:
 Two sides and the included angle.
 All three sides.
Example A
Solve the triangle using the Law of Sines. Round decimal answers to the nearest tenth.
First, to find
Now, use the Law of Sines to set up ratios for
Example B
Solve the triangle using the Law of Sines. Round decimal answers to the nearest tenth.
Set up the ratio for
To find
To find
Example C
Solve the triangle using Law of Cosines. Round your answers to the nearest hundredth.
Use the second equation to solve for
To find
Watch this video for help with the Examples above.
CK12 Foundation: Chapter8LawsofSinesandCosinesB
Vocabulary
The Law of Sines says
Guided Practice
Find the following angles in the triangle bloew. Round your answers to the nearest hundredth.
1.
2.
3.
Answers:
1. When you are given only the sides, you have to use the Law of Cosines to find one angle and then you can use the Law of Sines to find another.
2. Now that we have an angle and its opposite side, we can use the Law of Sines.
3. To find
Practice
Use the Law of Sines or Cosines to solve

m∠A=74∘,m∠B=11∘,BC=16 
m∠A=64∘,AB=29,AC=34 
m∠C=133∘,m∠B=25∘,AB=48
Use the Law of Sines to solve

m∠A=20∘,AB=12,BC=5
Recall that when we learned how to prove that triangles were congruent we determined that SSA (two sides and an angle not included) did not determine a unique triangle. When we are using the Law of Sines to solve a triangle and we are given two sides and the angle not included, we may have two possible triangles. Problem 14 illustrates this.
 Let’s say we have
△ABC as we did in problem 13. In problem 13 you were given two sides and the not included angle. This time, you have two angles and the side between them (ASA). Solve the triangle given thatm∠A=20∘,m∠C=125∘,AC=8.4  Does the triangle that you found in problem 14 meet the requirements of the given information in problem 13? How are the two different
m∠C related? Draw the two possible triangles overlapping to visualize this relationship.
Image Attributions
Description
Learning Objectives
Here you'll learn how to solve for missing sides and angles in nonright triangles using the Laws of Sines and Cosines.