1. Warm UP! Solve for all missing angles and sides: x 3 5 Y Z

2. What formulas did you use to solve the triangle? Pythagorean theorem SOHCAHTOA All angles add up to 180 o in a triangle

3. Could you use those formulas on this triangle? Solve for all missing angles and sides: 35 o 3 5 y z x This is an oblique triangle. An oblique triangle is any non-right triangle. There are formulas to solve oblique triangles just like there are for right triangles!

4. Solving Oblique Triangles Laws of Sines and Cosines & Triangle Area Students will solve trigonometric equations both graphically and algebraically.. Apply the law of sines and the law of cosines.

5. General Comments You have learned to solve right triangles. Now we will solve oblique triangles (non-right triangles). Note: Angles are Capital letters and the side opposite is the same letter in lower case. A B C a b c A C B a b c

6. What we already know The interior angles total 180. We can’t use the Pythagorean Theorem. Why not? For later, area = ½ bh Larger angles are across from longer sides and vice versa. The sum of two smaller sides must be greater than the third. A B C a b c

7. The Law of Sines helps you solve for sides or angles in an oblique triangle. (You can also use it upside-down)

8. Use Law of SINES when... AAS - 2 angles and 1 adjacent side ASA - 2 angles and their included side ASS – (SOMETIMES) 2 sides and their adjacent angle …you have 3 parts of a triangle and you need to find the other 3 parts. They cannot be just ANY 3 dimensions though, or you won’t have enough info to solve the Law of Sines equation. Use the Law of Sines if you are given:

9. General Process for Law Of Sines 1.Except for the ASA triangle, you will always have enough information for 1 full fraction and half of another. Start with that to find a fourth piece of data. 2.Once you know 2 angles, you can subtract from 180 to find the 3 rd. 3.To avoid rounding error, use given data instead of computed data whenever possible.

10. Example 1 Solve this triangle: AC B 70° 80° 12 c b The angles in a ∆ total 180°, so solve for angle C. Set up the Law of Sines to find side b: Angle C = 30° Side b = 12.6 cm Side c = 6.4 cm

11. Example 2: Solve this triangle A B C a b c 45 50 =30 85 You’re given both pieces for sinA/a and part of sinB/b, so we start there.

12. Example 3: Solve this triangle A B C a b c 3510 45 Since we can’t start one of the fractions, we’ll start by finding C. C = 180 – 35 – 10 = 135 135 Since the angles were exact, this isn’t a rounded value. We use sinC/c as our starting fraction. Using your calculator 36.5 11.1

13. You try! Solve this triangle AC B 115° 30° a = 30 c b

14. Example 3-Application A forest ranger at an observation point (A) sights a fire in the direction 32° east of north. Another ranger at a second observation point (B), 10 miles due east of A, sights the same fire 48° west of north. Find the distance from each observation point to the fire. 10 32 o 48 o 58 o 42 o A B 80 o 8.611 6.795

15. Example A civil engineer wants to determine the distances from points A and B to an inaccessible point C, as shown. From direct measurement, the engineer knows that AB = 25m,  A = 110 o, and  B = 20 o. Find AC and BC. A civil engineer wants to determine the distances from points A and B to an inaccessible point C, as shown. From direct measurement, the engineer knows that AB = 25m,  A = 110 o, and  B = 20 o. Find AC and BC. A B C

16. When solving an oblique triangle, using one of three available equations utilizing the cosine of an angle is handy. The equations are as follows: The Law of Cosines

17. The formula for the Law of Cosines makes use of three sides and the angle opposite one of those sides. We can use the Law of Cosines: SAS - two sides and the included angle SSS - all three sides General Strategies for Using the Law of Cosines

18. 87.0° 15.017.0 c B A Example 1: Solve this triangle Use the relationship: c 2 = a 2 + b 2 – 2ab cos C c 2 = 15 2 + 17 2 – 2(15)(17)cos(87°) c 2 = 487.309… c = 22.1 Now, since we know the measure of one angle and the length of the side opposite it, we can use the Law of Sines.

19. Example 2: Solve this triangle 31.4 23.2 38.6 C We start by finding cos A.

20. You TRY: 1.Solve a triangle with a = 8, b =10, and c = 12. 2.Solve a triangle with A = 88 o, B =16 o, and c = 14. A = 41.4 o a = 8 B = 55.8 o b = 10 C = 82.8 o c = 12 A = 88 o a = 12.4 B = 16 o b = 3.4 C = 76 o c = 14

21. IMPORTANT IT IS ALWAYS BEST TO USE LAW OF SINES FOR SIDES AND LAW OF COSINES FOR ANGLES Sometimes, however, it is just not possible – you may have to switch it up