1. The Law of Sines Section 6.1 Mr. Thompson

2. 2 An oblique triangle is a triangle that has no right angles. Definition: Oblique Triangles To solve an oblique triangle, you need to know the measure of at least one side and the measures of any other two parts of the triangle – two sides, two angles, or one angle and one side. C BA a b c

3. 3 The following cases are considered when solving oblique triangles. Solving Oblique Triangles 1.Two angles and any side (AAS or ASA) 2. Two sides and an angle opposite one of them (SSA) 3. Three sides (SSS) 4. Two sides and their included angle (SAS) A C c A B c a c b C c a c a B

4. The first two cases can be solved using the Law of Sines. (The last two cases can be solved using the Law of Cosines.) Definition: Law of Sines Law of Sines If ABC is an oblique triangle with sides a, b, and c, then Acute Triangle C BA b h c a C B A b h c a Obtuse Triangle

5. Find the remaining angle and sides of the triangle. Example: Law of Sines - ASA Example (ASA): The third angle in the triangle is A = 180  – A – B = 180  – 10  – 60  = 110 . C B A b c 60  10  a = 4.5 ft 110  Use the Law of Sines to find side b and c. 4.15 ft 0.83 ft

6. For the triangle in the figure, C = 102.3°, B = 28.7°, and b = 27.4 feet. Find the remaining angle and sides.

7. Ex. 2. A pole tilts toward the sun at an 8° angle from the vertical, and it casts a 22-foot shadow. The angle of elevation from the tip of the shadow to the top of the pole is 43°. How tall is the pole?

8. Remember back to Geometry - Which cases were enough to prove two triangles congruent? -Why was there a problem with Some of the cases? Try to come up with counterexamples for these cases

9. The Ambiguous Case (SSA) Table Knowing two sides and an angle opposite one of these sides might not be enough to form a triangle. This information may allow you to form 0, 1, or 2 triangles!

10. Use the Law of Sines to solve the triangle. A = 110 , a = 125 inches, b = 100 inches Example: Single Solution Case - SSA Example (SSA): C  180  – 110  – 48.74  C B A b = 100 in c a = 125 in 110  48.74  21.26  48.23 in = 21.26 

11. Use the Law of Sines to solve the triangle. A = 76 , a = 18 inches, b = 20 inches Example: No-Solution Case - SSA Example (SSA): There is no angle whose sine is 1.078. There is no triangle satisfying the given conditions. C A B b = 20 in a = 18 in 76 

12. Show that there is no triangle for which a = 15, b = 25, and A = 85°.

13. Use the Law of Sines to solve the triangle. A = 58 , a = 11.4 cm, b = 12.8 cm Example: Two-Solution Case - SSA Example (SSA): 72.2  10.3 cm Two different triangles can be formed. 49.8  a = 11.4 cm C A B1B1 b = 12.8 cm c 58  Example continues. C  180  – 58  – 72.2  = 49.8 

14. Use the Law of Sines to solve the second triangle. A = 58 , a = 11.4 cm, b = 12.8 cm Example: Two-Solution Case – SSA continued Example (SSA) continued: B 2  180  – 72.2  = 107.8  107.8  C A B2B2 b = 12.8 cm c a = 11.4 cm 58  14.2  3.3 cm 72.2  10.3 cm 49.8  a = 11.4 cm C A B1B1 b = 12.8 cm c 58  C  180  – 58  – 107.8  = 14.2 

15. Find two triangles for which a = 12 meters, b = 31 meters, and A = 20.5°.

16. My Old House Art Museum City Hall 1.25 miles

17. Area of an Oblique Triangle C BA b c a Find the area of the triangle. A = 74 , b = 103 inches, c = 58 inches Example: 74  103 in 58 in

18. 18 The following cases are considered when solving oblique triangles. Solving Oblique Triangles 1.Two angles and any side (AAS or ASA) 2. Two sides and an angle opposite one of them (SSA) 3. Three sides (SSS) 4. Two sides and their included angle (SAS) A C c A B c a c b C c a c a B

19. 19 The last two cases (SSS and SAS) can be solved using the Law of Cosines. (The first two cases can be solved using the Law of Sines.) Definition: Law of Cosines Law of Cosines Standard FormAlternative Form

20. 20 Find the three angles of the triangle. Example: Law of Cosines - SSS Example: C BA 8 6 12 Find the angle opposite the longest side first. Law of Sines: 36.3  117.3  26.4 

21. 21 Solve the triangle. Example: Law of Cosines - SAS Example: 67.8  Law of Sines: 37.2  C BA 6.2 75  9.5 9.9 Law of Cosines:

22. Example 1: Find the three angles of the triangle shown. 22

23. Example 2: Find the remaining angles and sides of the triangle shown. 23

24. 24 Definition: Heron’s Area Formula Heron’s Area Formula Given any triangle with sides of lengths a, b, and c, the area of the triangle is given by Example: Find the area of the triangle. 5 10 8

25. 25 Application: Law of Cosines Application: Two ships leave a port at 9 A.M. One travels at a bearing of N 53  W at 12 mph, and the other travels at a bearing of S 67  W at 16 mph. How far apart will the ships be at noon? 53  67  c 36 mi 48 mi C At noon, the ships have traveled for 3 hours. Angle C = 180  – 53  – 67  = 60  The ships will be approximately 43 miles apart. 43 mi 60  N

26. Example 5: Find the area of a triangle having sides a = 43 meters, b = 53 meters, and c = 72 meters. 26