1. C H. 6 – A DDITIONAL T OPICS IN T RIGONOMETRY 6.1 – Law of Sines

2. In this chapter, ΔABC has vertices A, B, and C, and the sides opposite those vertices are a, b, and c, respectively. Law of Sines: This law can be used to find all angles and sides of an oblique (not right) triangle You must know 3 angle or side measures to use this law, and 2 of the knowns must be an opposite side/angle pair Recall: Angles in a Δ sum to 180°! A B C c b a

3. Ex: Solve the triangle. To find A, subtract angles from 180°! 180 – 102.3 – 28.7 = 49° ! To find c, use Law of Sines! Hint: Use the given info instead of determined info when possible in case you made a mistake! Hint: Don’t evaluate or round until you have the answer! Hint: We’re in degrees! To find a, use Law of Sines! A B C c 27.4 ft a 28.7 ° 102.3 °

4. Ex: Solve the triangle. Can’t find anything but A right now! To find A, use Law of Sines! Cross-multiply to get… To find C, subtract angles from 180°! 180 – 42 – 21.41 = 116.59° ! To find c, use Law of Sines! To get the best answer, try not to round until the end! A B C c 22 in 12 in 42 °

5. S PECIAL C ASE #1 Ex: Solve the triangle. Can’t find anything but A right now! To find A, use Law of Sines! Cross-multiply to get… ERR: DOMAIN! Your calculator is telling you that there is no angle measure for A that will create a triangle. Why? Because in reality, 11 inches is not long enough to reach down to the 3 rd side. Final answer: NO SOLUTION ! A B C c 11 in 31 in 75 °

6. S PECIAL C ASE #2 Ex: Solve the triangle. To find A, use Law of Sines! Cross-multiply to get… BUT WAIT! You just remembered that there is a 2 nd quadrant angle with the same sine, mainly 180-A = 125.22 °! Therefore, there are 2 separate triangles that exist with the given measurements! You must solve both triangles! A B C c 29 in 46 in 31 °

7. S PECIAL C ASE #2 ( CONT ’ D ) Triangle #1: A = 54.78 ° To find C, subtract the angles from 180! 180 – 31 – 54.78 = 94.21° ! To find c, use Law of Sines! A B C c 29 in 46 in 31 °

8. S PECIAL C ASE #2 ( CONT ’ D ) Triangle #2: A = 125.22 ° To find C, subtract the angles from 180! 180 – 31 – 125.22 = 23.78° ! To find c, use Law of Sines! When do we know if there will be 2 triangles formed? 1. Only in the ASS condition 2. Only when the first angle found is greater than the given angle Basically, consider a 2 nd angle whenever you do sin -1 ! A B C c 29 in 46 in 31 °

9. A REA OF A TRIANGLE Area A B C c b a