1. Section 4.7: Law of Sines and Law of Cosines
Objectives:
-Solve oblique triangles using Law of Sines and Law of Cosines
-Find areas of oblique triangles.

2. Oblique Triangles
-Triangles that are not right triangles

3. Law of Sines When the 3 measurements provided fit one of these cases:
Two angles and a nonincluded side (AAS)
Two angles and the included side (ASA)
Two sides and the included angle (SSA) *
*choose any 2 ratios to create a proportion
with 3 known measurements, and 1 unknown.

4. Example 1
A) Solve ΔLMN. Round side lengths to the nearest tenth and angle measures to the nearest degree.

5. Example 1
B) Solve for y in ΔXYZ. Round side lengths to the nearest tenth.

6. Example 2
A) The angle of elevation from the top of a building to a hot air balloon is 62º. The angle of elevation to the hot air balloon from the top of a second building that is 650 feet due east is 49º. Find the distance from the hot air balloon to each building.

7. Example 2
B) A tree is leaning 10° past vertical as shown in the figure. A wire that makes a 42° angle with the ground 10 feet from the base of the tree is attached to the top of the tree. How tall is the tree?

8. Geometry Review
You know that the measures of two sides and a nonincluded angle (SSA) do not necessarily define a unique triangle. Consider the angle and side measures given in the figures below.
In general, given an SSA case, one of the following will be true:
No triangle exists (no solution)
Exactly 1 triangle exists (one solution)
Two triangles exist (two solutions)

9. **SOLVING SSA Triangles**
It is possible for more than 1 triangle to exist, or NO triangle to exist. It all depends on if the given angle is acute or obtuse.
Always look for 2 triangles when finding an angle using Law of Sines.
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10. Ambiguous Case SSA Given an Obtuse  Given an Acute 
*if OPP  ADJ, then NO ∆ exists *if ADJsin() > OPP, then NO ∆ exists
*if OPP > ADJ, then ONE ∆ exists *if ADJsin() = OPP, then ONE ∆ exists
*if OPP > ADJ, then ONE ∆ exists
*if ADJsin() < OPP < ADJ, then TWO ∆s exist

11. How to find the answers to the 2nd ∆:
1. Using the FIRST angle you found, subtract this number from 180º. This is the 2nd degree measure for the same angle.
2. The ORIGINAL angle given in the problem, will never change, so take this angle AND the angle from step 1 and subtract them from 180º. (Since all 3 angles have to add up to 180º.)
3. Use Law of Sines with your new angles to find the remaining side.

12. Example 3:
A) Find all solutions for the given triangle, if possible. If no solution exists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree. mA = 63°, a = 18, b = 25

13. Example 3:
B) Find all solutions for the given triangle, if possible. If no solution exists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree. mC = 105°, b = 55, c = 73,

14. Example 4:
A) Find all solutions for the given triangle, if possible. If no solution exists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree. mB = 45°, b = 18, and c = 24.

15. Example 4:
B) Find all solutions for the given triangle, if possible. If no solution exists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree. mC = 24°, c = 13, and a = 15.

16. Law of Cosines
When the 3 measurements provided fit one of these cases:
Three sides (SSS)
Two sides and the included angle (SAS)

17. Example 5:
A) A triangular area of lawn has a sprinkler located at each vertex. If the sides of the lawn are a = 19 feet, b = 24.3 feet, and c = 21.8 feet, what angle of sweep should each sprinkler be set to cover?

18. Example 5:
B) A triangular lot has sides of 120 feet, 186 feet, and 147 feet. Find the angle across from the shortest side.

19. Solving a  using Law of Cosines
*In SSS, you must find the LARGEST angle first,
then use the Law of Sines to find the SMALLER of the
2 remaining angles.
*In SAS, you must first find the side across from the known angle,
then use the Law of Sines to find the SMALLER of the
2 remaining angles.

20. Example 6:
A) Solve ΔABC. Round side lengths to the nearest tenth and angle measures to the nearest degree.

21. Example 6:
B) Solve ΔMNP if mM = 54o, n = 17, and p = 12. Round side lengths to the nearest tenth and angle measures to the nearest degree.

22. Example 6:
C) Solve ΔABC. Round angle measures to the nearest degree.

23. Area of Oblique Triangles
*Use for SSS*

24. Example 7:
A) Find the area of ΔABC to the nearest tenth.

25. Example 7:
B) Find the area of ΔGHJ to the nearest tenth.

26. Area of Oblique Triangles

27. Example 8:
A) Find the area of ΔABC to the nearest tenth.

28. Example 8:
B) Find the area of ΔDEF to the nearest tenth.