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Laws of Sines and Cosines

Solving for parts of non-right triangles using trigonometry.

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Laws of Sines and Cosines

What if you wanted to solve for a missing side in a non-right 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

CK-12 Foundation: Chapter8LawsofSinesandCosinesA

James Sousa: The Law of Sines

James Sousa: The Law of Cosines

Guidance

Law of Sines: If has sides of length, , and , then .

Looking at a triangle, the lengths , and are opposite the angles of the same letter.

Use Law of Sines when given:

• An angle and its opposite side.
• Any two angles and one side.
• Two sides and the non-included angle.

Law of Cosines: If has sides of length , and , then:

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 , we can use the Triangle Sum Theorem.

Now, use the Law of Sines to set up ratios for and .

Example B

Solve the triangle using the Law of Sines. Round decimal answers to the nearest tenth.

Set up the ratio for using Law of Sines.

To find use the Triangle Sum Theorem.

To find , use the Law of Sines again.

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 or , you can use either the Law of Sines or Law of Cosines. Let’s use the Law of Sines.

To find , use the Triangle Sum Theorem.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter8LawsofSinesandCosinesB

Guided Practice

Find the following angles in the triangle bloew. Round your answers to the nearest hundredth.

1.

2.

3.

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 , use the Triangle Sum Theorem.

Explore More

Use the Law of Sines or Cosines to solve . If you are not given a picture, draw one. Round all decimal answers to the nearest tenth.

Use the Law of Sines to solve below.

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.

1. Let’s say we have 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 that
2. Does the triangle that you found in problem 14 meet the requirements of the given information in problem 13? How are the two different related? Draw the two possible triangles overlapping to visualize this relationship.

To view the Explore More answers, open this PDF file and look for section 8.10.