Stitz-Zeager_College_Algebra_e-book

# Sin c y 3 2 2 7 1 5 6 6 1 3 1 1 3 x 1 x 2 1 x 1 3 1 o

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Unformatted text preview: 8.1 square units. The area of the triangle from Exercise 1m is about 377.1 square units. The area of the triangle from Exercise 1t is 149 square units. 11.3 The Law of Cosines 11.3 773 The Law of Cosines In Section 11.2, we developed the Law of Sines (Theorem 11.2) to enable us to solve triangles in the ‘Angle-Angle-Side’ (AAS), the ‘Angle-Side-Angle’ (ASA) and the ambiguous ‘Angle-Side-Side’ (ASS) cases. In this section, we develop the Law of Cosines which readily handles solving triangles in the ‘Side-Angle-Side’ (SAS) and ‘Side-Side-Side’ cases.1 We state and prove the theorem below. Theorem 11.5. Law of Cosines: Given a triangle with angle-side opposite pairs (α, a), (β, b) and (γ, c), the following equations hold a2 = b2 + c2 − 2bc cos(α) b2 = a2 + c2 − 2ac cos(β ) c2 = a2 + b2 − 2ab cos(γ ) To prove the theorem, we consider a generic triangle with the vertex of angle α at the origin with side b positioned along the positive x-axis. B = (c cos(α), c sin(α)) c a α A = (0, 0) b C = (b, 0) From this set-up, we immediately ﬁnd that the coordinates of A and C are A(0, 0...
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