Objectives Use the Law of Cosines to solve triangles. Apply Area Formula to triangles.

Notes #1-4 1. a = 18, b = 40, m C = 82.5° 2. x = 18; y = 10; z = 9 Use the given measurements to solve ∆ABC (nearest tenth). 1. a = 18, b = 40, m C = 82.5° 2. x = 18; y = 10; z = 9 3. Two model planes take off from the same spot. The first plane travels 300 ft due west before landing and the second plane travels 170 ft southeast before landing. To the nearest foot, how far apart are the planes when they land? 4. An artist needs to know the area of a triangular piece of stained glass with sides measuring 9 cm, 7 cm, and 5 cm. What is the area to the nearest square centimeter?

In the previous lesson, you learned to solve triangles by using the Law of Sines. However, the Law of Sines cannot be used to solve triangles for which side-angle-side (SAS) or side-side-side (SSS) information is given. Instead, you must use the Law of Cosines.

Example 1: Using the Law of Cosines Use the given measurements to solve ∆ABC. Round to the nearest tenth. a = 8, b = 5, mC = 32.2° Step 1 Find the length of the third side. c2 = a2 + b2 – 2ab cos C Law of Cosines c2 = 82 + 52 – 2(8)(5) cos 32.2° Substitute. c2 ≈ 21.3 Use a calculator to simplify. c ≈ 4.6 Solve for the positive value of c.

Example 1 Continued Step 2 Find the measure of the smaller angle, B. Law of Sines Substitute. Solve for m B. Step 3 Find the third angle measure. mA  112.4°

Example 2: Using the Law of Cosines Use the given measurements to solve ∆ABC. Round to the nearest tenth. a = 8, b = 9, c = 7 Step 1 Find the measure of the largest angle, B. b2 = a2 + c2 – 2ac cos B Law of cosines 92 = 82 + 72 – 2(8)(7) cos B Substitute. cos B = 0.2857 Solve for cos B. m B = Cos-1 (0.2857) ≈ 73.4° Solve for m B.

Step 2 Find another angle measure Example 2 Continued Use the given measurements to solve ∆ABC (nearest tenth). Step 2 Find another angle measure 72 = 82 + 92 – 2(8)(9) cos C Substitute Law of Cosines cos C = 0.6667 Solve for cos C. m C = Cos-1 (0.6667) ≈ 48.2° Solve for m C. Step 3 Find the third angle measure. m A  58.4°

The largest angle of a triangle is the angle opposite the longest side. Remember! When using the LAW of COSINES, find the largest angle first. When using the LAW of SINES, find the largest angle last (using the triangle sum formula)

Example 3 The surface of a hotel swimming pool is shaped like a triangle with sides measuring 50 m, 28 m, and 30 m. What is the area of the pool’s surface to the nearest square meter? Find the measure of the largest angle, A. Law of Cosines 502 = 302 + 282 – 2(30)(28) cos A m A ≈ 119.0° Solve for m A. 

Notes Use the given measurements to solve ∆ABC. Round to the nearest tenth. 1. a = 18, b = 40, m C = 82.5° c ≈ 41.7; m A ≈ 25.4°; m B ≈ 72.1° 2. x = 18; y = 10; z = 9 m Z ≈ 142.6°; m Y ≈ 19.7°; m Z ≈ 17.7°

Notes #3-4 3. Two model planes take off from the same spot. The first plane travels 300 ft due west before landing and the second plane travels 170 ft southeast before landing. To the nearest foot, how far apart are the planes when they land? 437 ft 4. An artist needs to know the area of a triangular piece of stained glass with sides measuring 9 cm, 7 cm, and 5 cm. What is the area to the nearest square centimeter? 17 cm2