a 2 = b 2 + c 2 - 2 b c cos A These two sides are repeated. Objectives: Use the Law of Cosines to solve triangles. Apply Area Formula to triangles. It doesn’t matter which side is called a, b, and c, as long as the opposite side is uppercase A, B, and C. a 2 = b 2 + c 2 - 2 b c cos A This side is always opposite this angle. These two sides are repeated. *Note if A = 90, this term drops out (cos 90 = 0), and we have the normal Pythagorean theorem.

Use these to find missing Sides. Use these to find missing Angles.

Write the rule and find Angle A: (I do, you watch)

Write the rule and find Angle B: (Whiteboard)

Write the rule and find Angle C: (Whiteboard)

Find the measures indicated. Round to the nearest tenth. (I do, you watch) Use the Law of Cosines to set up an equation to solve for m∠C. m∠C ≈ 60.8° Instead of using the Law of Cosines, m∠B can be found using the Triangle Sum Theorem since two of the angles are now known. m∠B ≈ 39.2°

Find m.

Find b. b ≈ 6.8

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°