10.2 The Law of Sines

1. 10.2 The Law of Sines Objectives: Solve oblique triangles by using the Law of Sines. Use area formulas to find areas of triangles.

2. The Law of Sines

3. Solve the triangle. Ex. #1 Solve a Triangle with AAS Information

4. For all these triangles, side a is called the swinging side and side b is called the fixed side. Ambiguous Case: SSA

5. Ambiguous Case: SSA

6. Given a possible triangle ABC with b=8, c=5, and C=54°, find angle B. First we draw the triangle. Since the exact shape is unknown, we start with a baseline and estimate a 54° angle. Side c must be across from angle C, so we place side b = 8 adjacent to the angle and c = 5 across from it. Since the angle is unknown, we can attach side c to b, we just don’t know the angle in which they meet. Side c in this scenario is called the swinging side. Ex. #2 Solve a Triangle with SSA Information

7. Given a possible triangle ABC with b=8, c=5, and C=54°, find angle B. The swinging side can form any angle in a circular motion attached at point A. Depending on the height h, from A to the baseline, the swinging side could either form one triangle, two triangles, or no triangle. Ex. #2 Solve a Triangle with SSA Information

8. Given a possible triangle ABC with b=8, c=5, and C=54°, find angle B. To find the height h, we will use right triangle trigonometry: The height is longer than the swinging side, which cannot reach the baseline to form a triangle. Ex. #2 Solve a Triangle with SSA Information

9. Lighthouse B is 3 miles east of lighthouse A. Boat C leaves lighthouse B and sails in a straight line. At the moment that the boat is 5 miles from lighthouse B, an observer at lighthouse A notes that the angle determined by the boat, lighthouse A (the vertex), and lighthouse B is 65°. Approximately how far is the boat from lighthouse A at that moment? In this situation, the swinging side is now a = 5 as angle A is given. Because the swinging side is longer than the fixed side c = 3, it will automatically be longer than the height h and form exactly one triangle. h Ex. #3 Solve a Triangle with SSA Information

10. Approximately how far is the boat from lighthouse A at that moment? To find side b, we must first find angle B. In order to find angle B we must first find angle C. Ex. #3 Solve a Triangle with SSA Information

11. Approximately how far is the boat from lighthouse A at that moment? Ex. #3 Solve a Triangle with SSA Information

12. Solve triangle ABC when a = 4.8, b = 6, & A = 28°. Here we must again first find the height in order to see if the swinging side a = 4.8 will reach down and touch the baseline to form a triangle. Ex. #4 Solve a Triangle with SSA Information

13. Solve triangle ABC when a = 4.8, b = 6, & A = 28°. With a height of 2.8, the swinging side is long enough to touch the baseline, but instead of forming just one triangle, it forms two triangles. 2.8 This occurs whenever the swinging side is longer than the height but shorter than the fixed side. Ex. #4 Solve a Triangle with SSA Information

14. Solve triangle ABC when a = 4.8, b = 6, & A = 28°. Since two possible triangles are formed, both need solved. The small numbers next to the sides and angles indicate which triangle we are solving. Ex. #4 Solve a Triangle with SSA Information

15. Solve triangle ABC when a = 4.8, b = 6, & A = 28°. To solve the second triangle, we must first use a little geometry and our answer for angle B1 to find angle B2. Since both legs (formed by the swinging side) are the same length, they form an isosceles triangle. This means the two interior angles are congruent as well. Since we found B1 to be 35.93°, to find B2 we subtract it from 180°. Ex. #4 Solve a Triangle with SSA Information

16. Solve triangle ABC when a = 4.8, b = 6, & A = 28°. To find angle C2we simplify must subtract angles A and B2 from 180°. Finally we can solve for side c2. Ex. #4 Solve a Triangle with SSA Information

17. Two surveyors, standing at points A and B, are measuring a building. The surveyor at point A is 20 feet further away from the building than the surveyor at point B. The angle of elevation of the top of the building from point A is 58°, and the angle of elevation of the top of the building from point B is 72°. How tall is the building? In order to figure out the height of the building, we will first need to find side a. Ex. #5 Solve a Triangle with ASA Information

18. How tall is the building? To find side a,we will need the other angles in its triangle. 14° 108° Ex. #5 Solve a Triangle with ASA Information

19. How tall is the building? To find the height of the building h,we will use simple right triangle trigonometry. 14° 108° Ex. #5 Solve a Triangle with ASA Information

20. Caution! This formula can only be directly used when given SAS information. Area of a Triangle Given SAS

21. Find the area of the triangle shown in the figure below: Ex. #6 Find Area with SAS Information

22. Heron’s Formula: Use this formula when all sides of a triangle are known. The variable s represents the semi-perimeter which is half the perimeter of the triangle. Area of a Triangle Given SSS

23. Find the area of the triangle whose sides have lengths of 8, 10, and 14. First find the semi-perimeter: Then plug into Heron’s formula and simplify: Ex. #7 Find Area with SSS Information