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Unformatted text preview: 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 . This alternate form is useful for finding angles when three sides are known. Example 3 (SSS) Solve 4 ABC if a = 10 , b = 15 , c = 20 . a=10 b=15 c=20 Solution: We can compute the angles using the second form of the Law of Cosines. A = cos 1 15 2 + 20 2 10 2 2 · 15 · 20 ≈ 28.95502 ◦ ≈ 29.0 ◦ B = cos 1 10 2 + 20 2 15 2 2 · 10 · 20 ≈ 46.56746 ◦ ≈ 46.6 ◦ For the third angle, we can either use the Law of Cosines again: C = cos 1 10 2 + 15 2 20 2 2 · 10 · 15 ≈ 104.47751 ◦ ≈ 104.5 ◦ or we can use the fact that the three angles sum to 180 ◦ : C = 180 ◦ A B ≈ 104.47751 ◦ ≈ 104.5 ◦ Now you try: 3. Solve 4 ABC if a = 39 , b = 35.6 , and c = 29.9 . (Round all results to the nearest tenth.) 5 Example 4 (SAS) Solve 4 ABC if a = 10 , b = 15 , C = 75 ◦ . a=10 b=15 C=75 º Solution: We can compute the the remaining side using the first form of the Law of Cosines. Then we compute the remaining angles. c = q 10 2 + 15 2 2 ( 10 )( 15 ) cos 75 ◦ ≈ 15.72750 ≈ 15.7 A = cos 1 15 2 + c 2 10 2 2 · 15 · c ≈ 37.89116 ◦ ≈ 37.9 ◦ B = cos 1 10 2 + c 2 15 2 2 · 10 · c ≈ 67.10883 ◦ ≈ 67.1 ◦ or we can use the fact that the three angles sum to 180 ◦ : B = 180 ◦ A C ≈ 67.10883 ◦ ≈ 67.1 ◦ Now you try: 4. Solve 4 ABC if B = 43.4 ◦ , a = 32.4 , and c = 30.9 . (Round all results to the nearest tenth.) 6 Part II: Which Method? Directions: Which method, Right Triangle Trigonometry, the Law of Cosines, or the Law of Sines, would be best to use as the first step in solving triangle ABC under the given conditions and why? Example 3: B = 65 ◦ ,a = 6,b = 7 . Method: Law of Sines Why: We are given “SSA.” Example 4: a = 3,C = 90 ◦ ,c = 5 . Method: Right Triangle Trigonometry (i.e. the Pythagorean Theorem and the definitions of sine, cosine, and tangent) Why: Triangle ABC is a right triangle Now you try: 5. Assume the angles of a triangle are labeled with the capital letters A,B, and C, and the side opposite an angle is labeled with the corresponding lowercase letter a,b, or c. Which method , “Right Triangle Trigonometry”, the “Law of Cosines”, or the “Law of Sines”, would be best to use as the first step in solving triangle ABC under the given conditions? To explain why , choose one of “right triangle”, “SSS”, “SAS”, “SSA”, or “AAS”. Do not solve the triangles. (a) b = 30 , c = 47 , C = 52 ◦ (d) a = 28 , b = 43 , c = 53 Which Method? Why? Which Method? Why? (b) B = 37 ◦ , c = 41 , C = 33 ◦ (e) a = 42 , b = 45 , C = 161 ◦ Which Method? Why? Which Method? Why? (c) A = 54 ◦ , c = 29 , C = 90 ◦ (f) a = 45 , A = 90 ◦ , b = 32 Which Method? Why? Which Method? Why?...
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 Spring '08
 Mcginnis
 Algebra, Trigonometry, triangle, Sines

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