Lesson 7-7 Law of Cosines

5-Minute Check on Lesson 7-6 Transparency 7-7 Click the mouse button or press the Space Bar to display the answers. Find each measure given the measures of ∆RST. Round all side measurements to the nearest tenth and angles to the nearest degree.. 1.Find s, if m  R = 63°, m  S = 38°, and r = Find m  R, if m  S = 122°, s = 10.8, and r = 5.2. Solve ∆MNP described below. Round all side measurements to the nearest tenth and angles to the nearest degree. 3.m  M = 50°, if m  N = 32°, and m = n = 8.5, p = 10.8, and m  P = 110°. 5. Find the perimeter of quadrilateral ABCD to the nearest tenth. Standardized Test Practice: ACBD B ° m  P = 98°, n = 10.4, p = ° 54 ° 8 cm m  N = 48°, m  M = 22°, m = 4.4

Objectives Use the Law of Cosines to solve triangles Solve problems by using the Law of Cosines

Vocabulary None new

Law of Cosines Let ∆ABC be any triangle with a, b and c representing the measures of the sides opposite the angles with measures A, B, and C respectively. Then the following equations are true: A B C a b c a 2 = b 2 + c 2 – 2bc cos A b 2 = a 2 + c 2 – 2ac cos B c 2 = a 2 + b 2 – 2ab cos C Law of Cosines can be used to solve triangles when the Law of Sines cannot be used Case 1: measures of two sides and their included angle (SAS) Case 2: measures of all three sides (SSS)

Example 1 Use the Law of Cosines since the measures of two sides and the included angle are known. Simplify. Take the square root of each side. Law of Cosines Use a calculator. Answer:

Example 2 Answer:

Example 3 Law of Cosines Simplify. Solve for L. Use a calculator. Subtract 754 from each side. Divide each side by –270. Answer:

Example 4 Answer:

Example 5 Determine whether the Law of Sines or the Law of Cosines should be used first to solve Then solve Round angle measures to the nearest degree and side measures to the nearest tenth. Since we know the measures of two sides and the included angle, use the Law of Cosines. Take the square root of each side. Use a calculator. Law of Cosines

Example 5 cont Cross products Divide each side by Law of Sines Next, we can find If we decide to find we can use either the Law of Sines or the Law of Cosines to find this value. In this case, we will use the Law of Sines.

Example 5 cont Use the Angle Sum Theorem to find Angle Sum Theorem Subtract 168 from each side. Answer: Take the inverse of each side. Use a calculator.

Example 6 Determine whether the Law of Sines or the Law of Cosines should be used first to solve Then solve Round angle measures to the nearest degree and side measures to the nearest tenth. Answer:

Summary & Homework Summary: –The Law of Cosines can be used to solve non-right triangles when you know 1) the measures of two sides and the included angle (SAS) or 2) the measures of all three sides (SSS) Homework: –pg ; 11, 12, 19-21, 27-30, 42