1. Assignment Trig Ratios III Worksheets (Online) Challenge Problem: Find a formula for the area of a triangle given a, b, and.

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3. Example 1 Find the area of ΔABC.

4. Trigonometric Ratios III Objectives: 1.To derive the Laws of Sines and Cosines 2.To find all the angles and sides of any triangle using the Laws of Sines and Cosines

5. Investigation: Law of Sines oblique Trigonometry can be applied to non-right, or oblique, triangles. In that example, we used it to find an unknown height. Now, we’ll use it to find a missing side length of a non-right triangle.

6. Investigation: Law of Sines Step 1: Draw ΔABC as shown with height h. Notice that side a is opposite  A, b is opposite  B, and c is opposite  C.

7. Investigation: Law of Sines Step 2: Write an equation for h using sin(  A). Step 3: Write an equation for h using sin(  B).

8. Investigation: Law of Sines Step 4: Use substitution to combine the two previous equations and write your final equation as a proportion.

9. Investigation: Law of Sines Step 5: Write an equation for k using sin(  B). Step 6: Write an equation for k using sin(  C).

10. Investigation: Law of Sines Step 7: Use substitution to combine the two previous equations and write your final equation as a proportion.

11. Investigation: Law of Sines Step 8: Finally, use the transitive property to combine the proportion from S4 and S7. This is the Law of Sines.

12. Law of Sines If ΔABC has side lengths a, b, and c as shown, then Looking for an ANGLE: Looking for a SIDE:

13. Example 2 Find the length of side AC in Δ ABC.

14. Example 3 Find the length of side AC in Δ ABC.

15. Acute or Obtuse? The Law of Sines can also be used to find a missing angle measure, but only if you know that it is acute or obtuse. This is simply because SSA is not a congruence shortcut.

16. Example 4 Solve ΔABC.

17. Example 5 Find the indicated measure. 1. 2.

18. Example 6 Find the length of AC in acute triangle ABC.

19. Law of Sines As the previous example demonstrates, you cannot always use the Law of Sines for every triangle. You need either two sides and an angle or two angles and a side in the following configurations: ASAAASSSA

20. Law of Cosines To derive the Law of Cosines, we need an interesting diagram like the one shown. Click the picture for a Geometer’s Sketchpad Demonstration of this useful equation.

21. Law of Cosines If  ABC has sides of length a, b, and c as shown, then

22. Example 7 Simplify the equation below for C = 90°.

23. Example 8 Solve the equation below for C.

24. Example 9 Find the length of AC in acute triangle ABC.

25. Example 10 Solve ΔABC.

26. Pro Tip: SSS When using the Law of Cosines to find a missing angle (SSS), it’s a good idea to find the angle opposite the longest side first. This is just in case the angle turns out to be obtuse. Regardless of what type of angle this turns out to be, use the Law of Sines and the Triangle Sum Theorem to find the other two angles. 1: Use the Law of Cosines 2: Use the Law of Sines 3: Use the Triangle Sum

27. Example 11 Find the indicated measure. 1. 2.

28. Summary ASA AAS SSA Law of Sines SAS SSS Law of Cosines Given three pieces of any triangle, you can use the Law of Sines or the Law of Cosines to completely solve the triangle.

29. Assignment Trig Ratios III Worksheets (Online) Challenge Problem: Find a formula for the area of a triangle given a, b, and.