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6.2 The Law of Cosines

6.2 The Law of Cosines. Solving an SAS Triangle. The Law of Sines was good for ASA - two angles and the included side AAS - two angles and any side SSA - two sides and an opposite angle (being aware of possible ambiguity) Why would the Law of Sines not work for an SAS triangle?. 15.

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6.2 The Law of Cosines

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  1. 6.2 The Law of Cosines

  2. Solving an SAS Triangle • The Law of Sines was good for • ASA - two angles and the included side • AAS - two angles and any side • SSA - two sides and an opposite angle (being aware of possible ambiguity) • Why would the Law of Sines not work for an SAS triangle? 15 26° No side opposite from any angle to get the ratio 12.5

  3. 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

  4. LAW OF COSINES LAW OF COSINES Do you see a pattern? Use these to findmissing sides Use these to find missing angles

  5. Deriving the Law of Cosines C • Write an equationusing Pythagorean theorem for shaded triangle. b h a k c - k A B c

  6. 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).

  7. Solve a triangle where b = 1, c = 3 and A = 80° Draw a picture. This isSAS B 3 a Do we know an angle and side opposite it? No so we must use Law of Cosines. C 80 1 Hint: we will be solving for the side opposite the angle we know. minus 2 times the productof 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

  8. We'll label side a with the value we found. We now have all of the sides but how can we find an angle? B 3 19.23 2.99 80 C 80.77 Hint: We have an angle and a side opposite it. 1 B is easy to find since the sum of the angles is a triangle is 180° If you found C first

  9. Solve a triangle where a = 5, b = 8 and c = 9 Draw a picture. This isSSS 9 B 5 Do we know an angle and side opposite it? No, so we must use Law of Cosines. A 84.26 C 8 Let's use largest side to find largest angle first. minus 2 times the productof 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

  10. How can we find one of the remaining angles? Do we know an angle and side opposite it? B 9 62.18 5 33.56 84.26  A 8 Yes, so use Law of Sines.

  11. Try it on your own! #1 • Find the three angles of the triangle ABC if C 8 6 A B 12

  12. Try it on your own! #2 • Find the remaining angles and side of the triangle ABC if C 16 80 A B 12

  13. Wing Span C • The leading edge ofeach wing of theB-2 Stealth Bombermeasures 105.6 feetin length. The angle between the wing's leading edges is 109.05°. What is the wing span (the distance from A to C)? • Note these are the actual dimensions! A

  14. Wing Span C A

  15. H Dub • 6-2 Pg. 443 #2-16even, 17-22all, and 29

  16. More Practice #1

  17. More Practice #2

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