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The Law of Cosines

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**Let's consider types of triangles with the three pieces of information shown below.**

We can't use the Law of Sines on these because we don't have an angle and a side opposite it. We need another method for SAS and SSS triangles. SAS AAA You may have a side, an angle, and then another side You may have all three angles. AAA This case doesn't determine a triangle because similar triangles have the same angles and shape but "blown up" or "shrunk down" SSS You may have all three sides

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**a c b What is the coordinate here?**

Let's place a triangle on the rectangular coordinate system. (a cos , a sin ) (x, y) What is the coordinate here? Drop down a perpendicular line from this vertex and use right triangle trig to find it. a c y x b (b, 0) Now we'll use the distance formula to find c (use the 2 points shown on graph) square both sides and FOIL This = 1 factor out a2 This is the Law of Cosines rearrange terms

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**LAW OF COSINES LAW OF COSINES**

We could do the same thing if gamma was obtuse and we could repeat this process for each of the other sides. We end up with the following: LAW OF COSINES Use these to find missing sides LAW OF COSINES Use these to find missing angles

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Since the Law of Cosines is more involved than the Law of Sines, when you see a triangle to solve you first look to see if you have an angle (or can find one) and a side opposite it. You can do this for ASA, AAS and SSA. In these cases you'd solve using the Law of Sines. However, if the 3 pieces of info you know don't include an angle and side opposite it, you must use the Law of Cosines. These would be for SAS and SSS (remember you can't solve for AAA). Since the Law of Cosines is more involved than the Law of Sines, when you see a triangle to solve you first look to see if you have an angle (or can find one) and a side opposite it. You can do this for ASA, AAS and SSA. In these cases you'd solve using the Law of Sines. However, if the 3 pieces of info you know don't include an angle and side opposite it, you must use the Law of Cosines. These would be for SAS and SSS (remember you can't solve for AAA).

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**a = 2.99 Solve a triangle where b = 1, c = 3 and = 80° This is SAS **

Draw a picture. This is SAS 3 a Do we know an angle and side opposite it? No so we must use Law of Cosines. 80 1 Hint: we will be solving for the side opposite the angle we know. minus 2 times the product of those other sides times the cosine of the angle between those sides One side squared sum of each of the other sides squared Now punch buttons on your calculator to find a. It will be square root of right hand side. a = 2.99 CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction

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**We'll label side a with the value we found.**

We now have all of the sides but how can we find an angle? 3 19.2 2.99 80 80.8 Hint: We have an angle and a side opposite it. 1 is easy to find since the sum of the angles is a triangle is 180° NOTE: These answers are correct to 2 decimal places for sides and 1 for angles. They may differ with book slightly due to rounding. Keep the answer for in your calculator and use that for better accuracy.

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**Solve a triangle where a = 5, b = 8 and c = 9**

Draw a picture. This is SSS 9 5 Do we know an angle and side opposite it? No, so we must use Law of Cosines. 84.3 8 Let's use largest side to find largest angle first. minus 2 times the product of those other sides times the cosine of the angle between those sides One side squared sum of each of the other sides squared CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction

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**How can we find one of the remaining angles?**

Do we know an angle and side opposite it? 9 62.2 5 84.3 33.5 8 Yes, so use Law of Sines.

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Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. Shawna has kindly given permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar

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