Aim: What is the Law of Sines and what good is it, anyway?

1. Aim: What is the Law of Sines and what good is it, anyway? The length of each of the equal sides of an isosceles triangle is a and the measure of a base angle is 15o. Express the area of the triangle in terms of a. Do Now:

2. Deriving the Law of Sines The Law of Sines Used to find the measure of a side of a triangle when the measures of two angles and a side are known (a.a.s. or a.s.a.).

3. Finding a Length In ∆ABC, a = 10, mA = 30, and mB = 50. Find b to the nearest integer. Law of Sines solve proportion: bSin30 = 10sin50 b = 15.32088886 To nearest integer, b = 15

4. Law of Sines A t =21 105º 27º D T Model Problem In ∆DAT, mD = 27, mA = 105, and t = 21. Find d to the nearest integer. 48º Establish ratios based on problem: solve proportion: dSin48 = 21sin27 To nearest integer, d = 12 d = 12.82899398

5. Law of Sines Model Problem In ∆ABC, a = 12, sinA = 1/3, and sinC = 1/4. Find c. Establish ratios based on problem: Solve proportion: 1/3c = 3 c = 9

6. Law of Sines Model Problem In ∆ABC, mB = 30 and mA= 45. Find the ratio a : b. Establish ratios based on problem: Solve proportion:

7. Model Problem (con’t) In ∆ABC, mB = 30 and mA= 45. Find the ratio a : b. Simplify:

8. B c 8.7 56º b A C Model Problem In right triangle ABC, mC = 90 and mA= 56, and BC = 8.7. Find AB to the nearest tenth. c = AB Establish ratios based on problem: Solve proportion: To nearest tenth, d = 10.5

9. Regents Prep Triangle ABC is an isosceles triangle. Its base is 16.2 cm. and one base angle is 63o20’. Find the length of one of the congruent sides to the nearest hundredth of a cm. 17.97 cm

10. Model Problem A surveyor at point P sights two points X and Y that are on opposite sides of a lake. If P is 200 m. from X and 350 m. from T, and mXPY = 40, find the distance from X to Y to the nearest meter.

11. The Product Rule