Law of Sines

1. Law of Sines What You will learn: Use the Law of Sines to solve oblique triangles when you know two angles and one side (AAS or ASA). Use the Law of Sines to solve oblique triangles when you know two sides and the angle opposite one of them (SSA).

2. Introduction In this section, we will solve oblique triangles – triangles that have no right angles. As standard notation, the angles of a triangle are labeled A, B, and C, and their opposite sides are labeled a, b,and c. To solve an oblique triangle, we need to know the measure of at least one side and any two other measures of the triangle—either two sides, two angles, or one angle and one side.

3. Introduction The Law of Sines can also be written in the reciprocal form:

4. Given Two Angles and One Side – AAS For the triangle below C = 102, B = 29, and b= 28 feet. Find the remaining angle and sides. By the triangle angle-sum theorem, A = 49.

5. B a c A C b Law of Sines Try this: By the triangle angle-sum theorem, C = 70.

6. Let’s look at this: Example 1 Given a triangle, demonstrate using the Law of Sines that it is a valid triangle (numbers are rounded so they may be up to a tenth off): a = 5 A = 40o b = 7 B = 64.1o c = 7.55C = 75.9o Is it valid?? a = 5 A = 40o b = 7 B = 115.9o c = 3.175 C = 24.1o Is it valid?? ??? ??? , so YES , so YES Why does this work? In both cases a, b and A are the same (two sides and an angle) but they produced two different triangles. Why??

7. Here is what happened • What is the relationship between 64.1and 115.9? • Remember the sine of an angle in the first quadrant (acute: 0o – 90o) and second quadrant,(obtuse: 90o – 180o)are the same!

8. The Ambiguous Case (SSA)

9. The Ambiguous Case (SSA) In our first example we saw that two angles and one side determine a unique triangle. However, if two sides and one opposite angle are given, three possible situations can occur: (1) no such triangle exists, (2) one such triangle exists, or (3) two distinct triangles may satisfy the conditions.

10. The Ambiguous Case (SSA)

11. Example – Single-Solution Case—SSA In triangle ABC, a = 12 inches, b = 5 inches, and A = 31. Find the remaining side and angles. Now, you can determine that C 180 – 31 – 12.39 = 136.61. Then, the remaining side is One solution: a b B is acute.

12. Example – No-Solution Case—SSA In triangle ABC, a = 4 inches, b = 14 inches, and A = 60. Find the remaining side and angles. NO SOLUTION

13. Example – Two-Solution Case—SSA In triangle ABC, a = 4.5 inches, b = 5 inches, and A = 58. Find the remaining side and angles. Now, you can determine that C1 180 – 58 – 70.44 = 51.56 C2 180 – 58 – 109.56 = 12.44 One solution: a b Two solutions

14. Law of Cosines What You will learn: Use the Law of coSinesto solve oblique triangles when you know three sides (SSS). Use the Law of coSinesto solve oblique triangles when you know two sides and the included angle (SAS). • Always solve for the angle across from the longest side first!

15. Example – SSS In triangle ABC, a = 6, b = 8, and c = 12. Find the three angles.

16. Example – SAS In triangle ABC, A = 80, b = 16, and c = 12. Find the remaining side and angles.