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Plement 1 of the acute angled solution and is given

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plement 1 of the acute-angled solution and is given by: θ 2 = 180 - sin - 1 x 180 sin 1 - x . 1 An angle and its supplement add up to a straight angle ( 180 ). So: supp θ = 180 - θ . 2
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Example 1 (SSA) Solve 4 ABC if a = 30 , b = 50 , and B = 61 . c o a = 30 b = 50 C B A 61 Solution: Because a, b, and B are given, we can use the equation a sin A = b sin B to find A. Plugging in a = 30, b = 50, and B = 61 , we find 30 sin A = 50 sin 61 = sin A = 30 sin 61 50 The SIN - 1 key on the calculator shows that A 31.65288385 . Store this result in your calculator under the letter ‘A.’ Rounding to the nearest tenth, A 31.7 Since we know A and B, we can compute C = 180 - A - 61 87.34711615 . Store this result in your calculator under the letter ‘C.’ Rounding to the nearest tenth, C 87.3 . Now we can use c sin C = a sin A or c sin C = b sin B to find c. Let’s use the second one. Plugging in b = 50 and B = 61 and leaving C stored in the calculator, we find c sin C = 50 sin 61 = c = 50 sin C sin 61 57.1064353 Rounding to the nearest tenth, c 57.1 Caution: An SSA configuration may have no solutions, one solution, or two solutions. There is an obtuse angle A which solves the equation: sin A = 30 sin 61 50 , namely: A 180 - 31.65288385 = 148.34711615 . But this obtuse angle does not give a second solution since any two angles of a triangle must add up to less than 180 . Now you try: 1. Solve 4 ABC if b = 14.1 , c = 11.8 , and C = 52.8 . (Round all results to the nearest tenth.) 3
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Example 2 (AAS) Solve 4 ABC if A = 33 , B = 46 , b = 4 . c o 33 o B A C b = 4 a 46 Solution: Since A and B are given, we can compute C = 180 - 33 - 46 = 101 . So C = 101 . We still must find a and c. To find a, we can use a sin A = b sin B or a sin A = c sin C . Because c is not yet known, we’ll have to use the first equation. Plugging in A = 33 , B = 46 , and b = 4, we find a sin 33 = 4 sin 46 = a = 4 sin 33 sin 46 3.028549427 Rounding to the nearest tenth, a 3.0 . To find c, we have a choice of using c sin C = a sin A or c sin C = b sin B . To avoid using the result from the previous computation, let’s use the second one. Plugging in C = 101 , b = 4, and , B = 46 , we find c sin 101 = 4 sin 46 = c = 4 sin 101 sin 46 5.458489482 Rounding to the nearest tenth, c 5.5 . Now you try: 2. Solve 4 ABC if A = 61.1 , C = 31.5 , and c = 10.9 . (Round all results to the nearest tenth.) 4
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Law of Cosines The Law of Cosines is usually given in a form that generalizes the Pythagorean Theorem: c 2 = a 2 + b 2 - 2ab cos C . This form is useful for finding one side when the opposite angle and other sides are known. Solve this form for the cosine: cos C = a 2 + b 2 - c 2 2ab .
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