Section 6.2 The Law of Cosines

About the Law of Cosines Can be used when given two sides and the angle between them (SAS) or when given three sides (SSS) but no angles. No ambiguous case. Make sure your calculator is in DEGREE mode. Once you identify a matched pair (an angle and its opposite side), you can resume using the Law of Sines.

LAW OF COSINES SSS SAS If the known parts of the triangle are SAS OR SSS, use the Law of Cosines. The known parts of shown in red in the triangles above.

Example 1: Solve the triangle where a = 6, b = 9, and c = 4 (Give angle measures to the nearest degree.) This information gives SSS. We will need to use the Law of Cosines to find an angle. Find the largest angle first. The largest angle is opposite the longest side. b2 = a2 + c2 -2ac cosB 92 = 62 + 42 -2(6)(4) cosB b2 = a2 + c2 -2ac cosB B = 127⁰. (continued)

Example 1 continued You may use the Law of Cosine or the Law of Sine to find another angle. The Law of Sine requires less work! A ≈ 320 C ≈ 180⁰ - 127⁰ - 32⁰ ≈ 21⁰

Example 2 Solve the triangle with A = 60 degrees, b = 20, c = 30. Round sides to the nearest tenth and angles to the nearest degree.

Example 3 Two ships leave a harbor at the same time. One ship travels on a bearing of S12⁰W at 14 miles per hours. The other ship travels on a bearing of N75⁰E at 10 miles per hour. How far apart will the ships be after three hours: (Round to the nearest tenth of a mile.)

Finding Area of an Oblique Triangle when the 3 sides are known. Heron’s Formula: Example 4: Find the area of a triangle having side lengths of a = 6 meters, b = 16 meters, and c = 18 meters. Round to the nearest square meter. Find s first. It is half of the perimeter of the triangle. s = (1/2)(6 + 16 + 18) = 20. Now use s in Heron’s Formula. Area ≈ 47 m2

Example 5 A piece of commercial real estate is priced at $3.50 per square foot. Find the cost, to the nearest dollar, of a triangular lot measuring 240 feet by 300 feet by 420 feet.

Example 6 A baseball diamond has four bases forming a square whose sides measure 90 feet each. The pitcher’s mound is 60.5 feet from home plate on a line joining home plate and second base. Find the distance from the pitcher’s mound to first base. Round to the nearest tenth of a foot.