This alternate form is useful for finding angles when

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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
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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
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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? 7
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Part III: Applications 6. Master Chief is trapped on an island with a Scorpion Tank as his only means of defense.
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