Pre-Calc Exam Notes 42

Pre-Calc Exam Notes 42 - C is acute and for when C is...

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42 Chapter 2 General Triangles §2.1 b a A C B (a) a b : No solution b a A C B (b) a > b : One solution Figure 2.1.4 The ambiguous case when A 90 Table 2.1 summarizes the ambiguous case of solving ABC when given a , A , and b . Of course, the letters can be interchanged, e.g. replace a and A by c and C , etc. Table 2.1 Summary of the ambiguous case 0 < A < 90 90 A < 180 a < b sin A : No solution a b : No solution a = b sin A : One solution a > b : One solution b sin A < a < b : Two solutions a b : One solution There is an interesting geometric consequence of the Law of Sines. Recall from Section 1.1 that in a right triangle the hypotenuse is the largest side. Since a right angle is the largest angle in a right triangle, this means that the largest side is opposite the largest angle. What the Law of Sines does is generalize this to any triangle: In any triangle, the largest side is opposite the largest angle. To prove this, let C be the largest angle in a triangle ABC . If C = 90 then we already know that its opposite side c is the largest side. So we just need to prove the result for when
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Unformatted text preview: C is acute and for when C is obtuse. In both cases, we have A ≤ C and B ≤ C . We will Frst show that sin A ≤ sin C and sin B ≤ sin C . y x r r ( x 1 , y 1 ) ( x 2 , y 2 ) A C Figure 2.1.5 If C is acute, then A and B are also acute. Since A ≤ C , imagine that A is in standard position in the xy-coordinate plane and that we rotate the terminal side of A counterclockwise to the terminal side of the larger angle C , as in ±igure 2.1.5. If we pick points ( x 1 , y 1 ) and ( x 2 , y 2 ) on the terminal sides of A and C , respectively, so that their distance to the origin is the same number r , then we see from the picture that y 1 ≤ y 2 , and hence sin A = y 1 r ≤ y 2 r = sin C . By a similar argument, B ≤ C implies that sin B ≤ sin C . Thus, sin A ≤ sin C and sin B ≤ sin C when C is acute. We will now show that these inequalities hold when C is obtuse....
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