1. The Law of COSINES

2. Objectives: CCSS To find the area of any triangle. To use the Law of Cosine; Understand and apply. Derive the formula for Law of Cosines

3. The Law of COSINES For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles:

4. Example 1: Given SAS Find all the missing dimensions of triangle ABC, given that angle B = 98°, side a = 13 and side c = 20. B 98° C a = 13 A c = 20 b Use the Law of Cosines equation that uses a, c and B to find side b: Start with finding the side with the angle you have. Fill in missing known parts Solve for unknown What is the only angle you know?

5. Example 1: Given SAS teaching note Find all the missing dimensions of triangle ABC, given that angle B = 98°, side a = 13 and side c = 20. B 98° C a = 13 A c = 20 b Use the Law of Cosines equation that uses a, c and B to find side b:

6. Example 1: Given SAS Now that we know B and b, we can use the Law of Sines to find one of the missing angles: B 98° C a = 13 A c = 20 b = 25.3 What Law of Sines?

7. Example 1: Given SAS teaching note Now that we know B and b, we can use the Law of Sines to find one of the missing angles: B 98° C a = 13 A c = 20 b = 25.3 Solution: b = 25.3, C = 51.5°, A = 30.5°

8. Example 2: Given SAS Find all the missing dimensions of triangle, ABC, given that angle A = 39°, side b = 20 and side c = 15. Use the Law of Cosines equation that uses b, c and A to find side a: 39° A b = 20 c = 15 B C a That is the side you start finding What is the only angle you know?

9. Example 2: Given SAS Find all the missing dimensions of triangle, ABC, given that angle A = 39°, side b = 20 and side c = 15. Use the Law of Cosines equation that uses b, c and A to find side a: 39° A b = 20 c = 15 B C a

10. Example 2: Given SAS Use the Law of Sines to find one of the missing angles: 39° A b = 20 c = 15 B C a = 12.6 Important: Notice that we used the Law of Sine equation to find angle C rather than angle B. The Law of Sine equation will never produce an obtuse angle. If we had used the Law of Sine equation to find angle B we would have gotten 87.5°, which is not correct, it is the reference angle for the correct answer, 92.5°. If an angle might be obtuse, never use the Law of Sine equation to find it.

11. Example 2: Given SAS teaching note Use the Law of Sines to find one of the missing angles: 39° A b = 20 c = 15 B C a = 12.6 Important: Notice that we used the Law of Sine equation to find angle C rather than angle B. The Law of Sine equation will never produce an obtuse angle. If we had used the Law of Sine equation to find angle B we would have gotten 87.5°, which is not correct, it is the reference angle for the correct answer, 92.5°. If an angle might be obtuse, never use the Law of Sine equation to find it.

12. Example 3: Given SSS Find all the missing dimensions of triangle, ABC, given that side a = 30, side b = 20 and side c = 15. A C B a = 30 c = 15 b = 20 We can use any of the Law of Cosine equations, filling in a, b & c and solving for one angle. Once we have an angle, we can either use another Law of Cosine equation to find another angle, or use the Law of Sines to find another angle.

13. Example 3: Given SSS Important: The Law of Sines will never produce an obtuse angle. If an angle might be obtuse, never use the Law of Sines to find it. For this reason, we will use the Law of Cosines to find the largest angle first (in case it happens to be obtuse). A C B a = 30 c = 15 b = 20 Angle A is largest because side a is largest:

14. Example 3: Given SSS teaching note Important: The Law of Sines will never produce an obtuse angle. If an angle might be obtuse, never use the Law of Sines to find it. For this reason, we will use the Law of Cosines to find the largest angle first (in case it happens to be obtuse). A C B a = 30 c = 15 b = 20 Angle A is largest because side a is largest:

15. Example 3: Given SSS Use Law of Sines to find angle B or C (its safe because they cannot be obtuse): A C B a = 30 c = 15 b = 20 117.3°

16. Example 3: Given SSS teaching note Use Law of Sines to find angle B or C (its safe because they cannot be obtuse): A C B a = 30 c = 15 b = 20 117.3° Solution: A = 117.3° B = 36.3° C = 26.4°

17. The Law of Cosines SAS SSS When given one of these dimension combinations, use the Law of Cosines to find one missing dimension and then use Law of Sines to find the rest. Important: The Law of Sines will never produce an obtuse angle. If an angle might be obtuse, never use the Law of Sines to find it.