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Section 5.6 The Law of Sines Objective: Students will be able to: 1. Solve triangles using the Law of Sines if the measures of 2 angles and a side are given. 2. Find the area of a triangle if the measures of 2 sides and the included angle or the measures of 2 angles and a side are given. Law of Sines The ___________________ can be used to solve triangles that are not right triangles. Law of Sines Let βπ΄π΅πΆ be any triangle with a, b and c representing the measures of the sides opposite the angles with measures A, B, and C respectively. Then, the following is true: π π π = = sin π΄ sin π΅ sin πΆ From geometry, you know that a unique triangle can be formed if you know the measures of 2 angles and the included side (ASA) or the measures of 2 angles and the non-included side (AAS). Therefore, there is one unique solution when you use the Law of Sines to solve a triangle given the measures of 2 angles and one side. *** In Section 5.7, you will learn how to use the Law of Sines when the measures of 2 sides and a nonincluded angle are given. Example 1: Solve ο²ABC if A = 24°, B = 62°, and a = 21.4. First, find the measure of οC. C = 180° - (24° + 62°) or 94° Use the Law of Sines to find b and c. Find b : π sin π΄ π = sin π΅ Find c : π sin πΆ = π sin π΄ 21.4 π = sin 24° sin 62° π 21.4 = sin 94° sin 24° 21.4 sin 62° =π sin 24° π= 46.45531404 π β 46.5 b 21.4 sin 94° sin 24° 52.48573257 π β 52.5 c Example 2: The angle of depression from a window of a house to the front edge of the swimming pool is 26.6°. The angle of depression from this same window to the back edge of the swimming pool is 15.3°. The length of the pool is 25 feet. If a person looks out the window, about how far is he from the front edge of the pool? *** Make a diagram of the problem. Remember that the angle of elevation is congruent to the angle of depression because they are alternate interior angles. First, find ο±. ο± = 26.6° - 15.3° or 11.3° Use the Law of Sines to find d. π 25 = sin 15.3° sin 11.3° 25 sin 15.3° π= sin 11.3° d 33.66652748 The person would be about 33.7 feet from the front edge of the pool. The area of any triangle can be expressed in terms of 2 sides of a triangle and the measure of the included angle. Area of Triangles Let β π΄π΅πΆ be any triangle with a, b, and c representing the measures of the sides opposite the angles with measurements A, B, and C, respectively. Then the area K can be determined using one of the following formulas: πΎ= 1 ππ sin π΄ 2 , πΎ= 1 ππ sin π΅ 2 , πΎ= Example 3: Find the area of ο²ABC if b = 14.8, c = 10.2, and A = 54°12ο’. 1 K = ππ sin π΄ 2 1 2 πΎ = (14.8)(10.2) sin (54° 12β² ) K 61.21909706 The area of ο²ABC is about 61.2 square units. 1 ππ sin πΆ 2 You can also find the area of a triangle if you know the measures of one side and 2 π π π sin π΅ angles of the triangle. By the Laws of Sines , sin π΅ = sin πΆ ππ π = sin πΆ , if you substitute π sin π΅ sin πΆ 1 π for b in K = 2 ππ sin π΄, the result is K = π ππ π¬π’π§ π¨ π¬π’π§ π© π¬π’π§ πͺ . Two similar formulas can be developed from the Laws of Sines: π π² = π ππ π¬π’π§ π© π¬π’π§ πͺ π¬π’π§ π¨ and π π² = π ππ πππ π¨ πππ πͺ πππ π© Example 4: Find the area of ο²DEF if e = 18.6, E = 78.2°, and F = 41.3°. First find the measure of οD. D = 180° - (78.2° + 41.3°) or Then, find the area of the triangle. 1 sin π· sin πΉ K = π2 2 sin πΈ 1 sin 60.5° sin 41.3° πΎ = (18.6)2 2 sin 78.2° 60.5° K 101.5111719 The area of ο²DEF is about 101.5 square units.