Law of Cosines.ppt Law of Cosines.ppt Law of Cosines.ppt
- 1. Law of Cosines MELC: • Illustrates the laws of sines and cosines (M9GE-IV-f-g-1)
- 2. 2 Recall: Law of Sines The first two cases can be solved using the Law of Sines (AAS/ASA, SSA/Angle Side Side), whereas the last two cases can be solved using the Law of Cosines (SSS, SAS).
- 3. 3 What You Should Learn • Use the Law of Cosines to solve oblique triangles (SSS or SAS).
- 5. 5 Introduction Two cases remain in the list of conditions needed to solve an oblique triangle—SSS and SAS. To use the Law of Sines, you must know at least one side and its opposite angle. When you are given three sides (SSS), or two sides and their included angle (SAS), none of the ratios in the Law of Sines would be complete. In such cases you can use the Law of Cosines.
- 7. 7 Proving the Law of Cosines Consider triangle ABC with sides a, b, and c with an altitude x.
- 8. 8 Proving the Law of Cosines Consider triangle ABC with sides a, b, and c with an altitude x.
- 10. 10 Example 1 – Three Sides of a Triangle—SSS Find the three angles of the triangle shown in Figure 1. Solution: It is a good idea first to find the angle opposite the longest side—side b in this case. Using the alternative form of the Law of Cosines, you find that Figure 1 Alternative form
- 11. 11 Example 1 – Solution cont’d Substitute for a, b, and c. Simplify. Because cos B is negative, you know that B is an obtuse angle given by B ≈ 116.80 º
- 12. 12 Example 1 – Solution At this point it is simpler to use the Law of Sines to determine A. You know that A must be acute, because B is obtuse, and a triangle can have, at most, one obtuse angle. So, A 22.08 and cont’d
- 13. 13 Example 2 – Two Sides and their Included Angle—SAS Find the remaining sides and angles of the triangle shown in Figure 2. Figure 1