7.7 Law of Cosines

Objective Use the Law of Cosines to solve triangles and problems

Law of Cosines Previously, we learned the Law of Sines, which as some theorems can, it does have its limitations. To use the Law of Sines we had to know the measures of two angles and any side (AAS or ASA) OR the measures of two sides and an opposite angle (SSA). If we don’t have either of these scenarios, but instead we have the measures of two sides and the included angle (SAS) or the measures of all the sides and no angles (SSS), we must use the Law of Cosines. a b

Law of Cosines Law of Cosines: In ABC, a2 = b2 + c2 – 2bc cos A b2 = a2 + c2 – 2ac cos B c2 = a2 + b2 – 2ab cos C

Example 1: Use the Law of Cosines since the measures of two sides and the included angle are known.

Example 1: Law of Cosines Simplify. Take the square root of each side. Use a calculator. Answer:

Your Turn: Answer:

Example 2: Law of Cosines Simplify.

Example 2: Subtract 754 from each side. Divide each side by –270. Solve for L. Use a calculator. Answer:

Your Turn: Answer:

Example 3: 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.

Example 3: Law of Cosines Take the square root of each side. Use a calculator. 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 3: Law of Sines Cross products Divide each side by 46.9. Take the inverse of each side. Use a calculator.

Example 3: Use the Angle Sum Theorem to find Angle Sum Theorem Subtract 168 from each side. Answer:

Your Turn: 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:

Example 4: AIRCRAFT From the diagram of the plane shown, determine the approximate exterior perimeter of each wing. Round to the nearest tenth meter. Since is an isosceles triangle,

Example 4: Use the Law of Sines to find KJ. Law of Sines Cross products Divide each side by sin . Simplify.

Example 4: Use the Law of Sines to find . Law of Sines Cross products Divide each side by 9. Solve for H. Use a calculator.

Example 4: Use the Angle Sum Theorem to find Angle Sum Theorem Subtract 95 from each side.

Example 4: Use the Law of Sines to find HK. Law of Sines Cross products Divide each side by sin Use a calculator.

Example 4: The perimeter of the wing is equal to Answer: The perimeter is about or about 67.1 meters.

Your Turn: The rear side window of a station wagon has the shape shown in the figure. Find the perimeter of the window if the length of DB is 31 inches. Answer: about 93.5 in.

Assignment Pre-AP Geometry: Pg. 388 #12 – 38 evens, 40 & 42