Review

Section 6.6 The Law of Cosines

The Law of Sines cannot be used directly to solve triangles if we know two sides and the angle between them or if we know all three sides (these are Cases 3 and 4 of the preceding section). In these two cases, the Law of Cosines applies.

Proof: To prove the Law of Cosines, place triangle ABC so that ∠A is at the origin, as shown in the Figure on the right. The coordinates of the vertices B and C are (c, 0) and (b cos A, b sin A), respectively. Using the Distance Formula, we get a2 = (c − b cos A)2 + (b sin A − 0)2 = c2 − 2bc cos A + b2 cos2 A + b2 sin2 A = c2 − 2bc cos A + b2 (cos2 A + sin2 A) = b2 + c2 − 2bc cos A This proves the first formula. The other two formulas are obtained in the same way by placing each of the other vertices of the triangle at the origin and repeating the preceding argument.

EXAMPLE: A tunnel is to be built through a mountain. To estimate the length of the tunnel, a surveyor makes the measurements shown in the Figure on the right. Use the surveyor’s data to approximate the length of the tunnel. Solution: To approximate the length c of the tunnel, we use the Law of Cosines: c2 = a2 + b2 − 2ab cos C = 2122 + 3882 − 2(212)(388) cos 82.4◦ ≈ 173730.2367 √ c ≈ 173730.2367 ≈ 416.8 Thus, the tunnel will be approximately 417 ft long. EXAMPLE: The sides of a triangle are a = 5, b = 8, and c = 12. Find the angles of the triangle.

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EXAMPLE: The sides of a triangle are a = 5, b = 8, and c = 12. Find the angles of the triangle.

Solution: We first find ∠A. From the Law of Cosines, we have a2 = b2 + c2 − 2bc cos A. Solving for cos A, we get cos A = b 2 + c2 − a 2 82 + 122 − 52 183 = = = 0.953125 2bc 2(8)(12) 192

Using a calculator, we find that ∠A = cos−1 (0.953125) ≈ 18◦ . In the same way the equations cos B = a2 + c2 − b2 52 + 122 − 82 = = 0.875 2ac 2(5)(12) a2 + b2 − c2 52 + 82 − 122 = = −0.6875 2ab 2(5)(8)

cos C =

give ∠B ≈ 29◦ and ∠C ≈ 133◦ . Of course, once two angles are calculated, the third...