Equating the x y term to 0 we get 2a cos sin b cos2

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Unformatted text preview: Cosines to find the length of the missing side opposite the given angle. Calling this length w (for width ), we get w2 = 9502 + 10002 − 2(950)(1000) cos (60◦ ) = 952500 from which √ we get w = 952500 ≈ 976 feet. In Section 11.2, we used the proof of the Law of Sines to develop Theorem 11.4 as an alternate formula for the area enclosed by a triangle. In this section, we use the Law of Cosines to prove Heron’s Formula – a formula which computes the area enclosed by a triangle using only the lengths of its sides. Theorem 11.6. Heron’s Formula: Suppose a, b and c denote the lengths of the three sides of 1 a triangle. Let s be the semiperimeter of the triangle, that is, let s = 2 (a + b + c). Then the area A enclosed by the triangle is given by A= s(s − a)(s − b)(s − c) We begin proving Theorem 11.6 using Theorem 11.4. Using the convention that the angle γ is opposite the side c, we have A = 1 ab sin(γ ) from Theorem 11.4. In order to simplify computations, 2 we start...
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