1. 7/3/2015 10.4: The Law of Sines Expectation: G1.3.2: Know and use the Law of Sines and the Law of Cosines and use them to solve problems. Find the area of a triangle with sides a and b and included angle θ using the formula Area = (1/2) a b sin θ.

2. 7/3/2015 10.4: The Law of Sines The Law of Sines For any triangle Δ ABC, with sides of a, b, and c opposite ∠ A, ∠ B and ∠ C respectively:

3. 7/3/2015 10.4: The Law of Sines 3 Cases of The Law of Sines 1. Acute Triangles 2. Right Triangles 3. Obtuse Triangles

4. 7/3/2015 10.4: The Law of Sines Proving the Law of Sines Given: Δ ABC is acute Prove:Sin ASin BSin C a c b

5. 7/3/2015 10.4: The Law of Sines Acute Case: Law of Sines A B C a b c h x y

6. 7/3/2015 10.4: The Law of Sines Acute Case: Law of Sines A B C a b c g tu

7. 7/3/2015 10.4: The Law of Sines Solve the triangle below (calculate all unknown measures). 74° 20 15 A b B

8. 7/3/2015 10.4: The Law of Sines Solve the triangle below. 60°42° 5 A B C b a

9. In the figure below, what is the measure of ∠ α, rounded to the nearest whole degree? a. 20° b. 55° c. 70° d. 75° e. 110° 7/3/2015 The Law of Sines ∠α∠α 55° 10 11.47

10. 7/3/2015 10.4: The Law of Sines Two cars leave the same house at the same time and each is 40 miles away after 1 hour. Car A traveled in the direction of 60° north of east. Car B traveled in the direction of 25° west of north. How far apart (from each other) are the two cars?

11. 7/3/2015 10.4: The Law of Sines Solve the triangle below. C A 30° 7 B c 12

12. 7/3/2015 10.4: The Law of Sines Solve the triangle below. A B 7 30° C 12

13. 7/3/2015 10.4: The Law of Sines What Happened? The Ambiguous Case: A triangle with an angle and 2 nonincluded sides given (SSA) may have 2 different solutions.

14. 7/3/2015 10.4: The Law of Sines SsA Condition SSA is not a problem if the given angle is opposite the longer of the 2 given sides. This is called SsA. A triangle to be solved is not ambiguous if it satisfies SsA.

15. 7/3/2015 10.4: The Law of Sines SsA 5 cm 12 cm 110° This is not an ambiguous case!

16. 7/3/2015 10.4: The Law of Sines SsA 12 cm 5 cm 18° This is an ambiguous case!

17. 7/3/2015 10.4: The Law of Sines What to Do? What to Do? If the case is ambiguous, use sin α = sin (180 - α ) to determine both possible angle combinations.

18. 7/3/2015 10.4: The Law of Sines Solve the triangle below. 30° 7 C B A c 12

19. Yana is planning a triangular garden. He wants to put a fence around it. The length of one side is 30 feet. If the angles at each end of this side are 44° and 58°, find the length of fencing needed to enclose the garden. 7/3/2015 8-5: The Law of Sines

20. 7/3/2015 10.4: The Law of Sines Assignment: Pages 658 – 661, # 13-21 (odds), 33-35, 51-56 (all).