Precalculus PreAP/Dual, Revised ©2017 Law of Cosines Section 6.2 Precalculus PreAP/Dual, Revised ©2017 Viet.dang@humbleisd.net 11/17/2018 9:48 AM §6.2: Law of Cosines

Review Find the missing side. 11/17/2018 9:48 AM §6.2: Law of Cosines

Review Find the missing side. 11/17/2018 9:48 AM §6.2: Law of Cosines

Steps Draw the figure and label the missing parts of the shape Plug in Law of Cosines equation, 𝒂 𝟐 = 𝒃 𝟐 + 𝒄 𝟐 −𝟐𝒃𝒄 𝐜𝐨𝐬𝑨 Make sure the answer is in degree mode Round answers to 4 decimal places Check 11/17/2018 9:48 AM §6.2: Law of Cosines

Law of Cosines Used for triangles that are not right triangles (AKA Oblique Triangles) Vertices are represented by a capital letter Sides are represented by a lower case letter Used to solve triangles for which side-angle-side (SAS) or side-side-side (SSS) 11/17/2018 9:48 AM §6.2: Law of Cosines

Law of Cosines Equation to Find Sides Angle is Capitalized Sides are Lower Cased 11/17/2018 9:48 AM §6.2: Law of Cosines

Proof Consider the following triangle, ABC. From the Right Triangle ADC, use the Pythagorean Theorem 11/17/2018 9:48 AM §6.2: Law of Cosines

Example 1 Solve for side 𝒃 when 𝒂=𝟖, 𝒄=𝟓, and 𝑩=𝟕𝟕 11/17/2018 9:48 AM §6.2: Law of Cosines

Your Turn Solve for side 𝒄 when 𝒂=𝟖, 𝒃=𝟓, and 𝑪=𝟑𝟐.𝟐 11/17/2018 9:48 AM §6.2: Law of Cosines

Example 2 Solve for angle 𝑹 when 𝒓=𝟗𝟎, 𝒔=𝟕𝟎, and 𝒕=𝟒𝟎.𝟓 §6.2: Law of Cosines

Example 3 Solve for angle 𝑻 when 𝒓=𝟗𝟎, 𝒔=𝟕𝟎, and 𝒕=𝟒𝟎.𝟓 §6.2: Law of Cosines

Your Turn Solve for angle 𝑩 when 𝒂=𝟕, 𝒃=𝟑, and 𝒄=𝟓 11/17/2018 9:48 AM §6.2: Law of Cosines

Example 4 A ship 𝑹 leaves port 𝑷 at a rate of 𝟏𝟔 miles per hour. Another ship 𝑸 leaves the same port at the same time and travels 𝟏𝟐 miles per hour. Both ships are traveling in straight paths. The direction taken by the two ships makes an angle of 𝟓𝟓°. How far apart are the ships after 𝟑 hours? 11/17/2018 9:48 AM §6.2: Law of Cosines

Example 4 A ship 𝑹 leaves port 𝑷 at a rate of 𝟏𝟔 miles per hour. Another ship 𝑸 leaves the same port at the same time and travels 𝟏𝟐 miles per hour. Both ships are traveling in straight paths. The direction taken by the two ships makes an angle of 𝟓𝟓°. How far apart are the ships after 𝟑 hours? 11/17/2018 9:48 AM §6.2: Law of Cosines

Example 5 A pilot is flying from Houston to Oklahoma City. The usual straight distance to OKC is 𝟑𝟗𝟔 miles. To avoid a thunderstorm, the pilot flies 𝟐𝟖° off the direct route for a distance of 𝟏𝟕𝟓 miles. He then makes a turn and flies straight on to Oklahoma City. How much farther than the direct route was the route taken by the pilot? 11/17/2018 9:48 AM §6.2: Law of Cosines

Your Turn The pitcher’s mound on a women’s softball field is 𝟒𝟑 feet from home plate and the distance between the bases is 𝟔𝟎 feet. (The pitcher’s mound is not halfway between home plate and second base.) The angle from home plate to pitcher’s mound to first base is 𝟒𝟓°. How far is the pitcher’s mound from first base? 11/17/2018 9:48 AM §6.2: Law of Cosines

About Heron Determine the area of this triangle in centimeters Physicist, mathematician, and engineer Taught at the museum in Alexandria Interests were more practical (mechanics, engineering, measurement) than theoretical He is placed somewhere around 75 A.D. (±150) Discovered areas of quadrilaterals, polygons from 3-12 sides, and circle and surface areas of 3-D objects Heron’s Aeolipile was the first recorded steam engine. It was taken as being a toy but could have possibly caused an industrial revolution 2000 years before the original. 11/17/2018 9:48 AM §6.2: Law of Cosines

Proof Using Euclid’s Elements of Geometry: The opposite angles of a cyclic quadrilateral sum to two right angles all derived from the Law of Cosines 11/17/2018 9:48 AM §6.2: Law of Cosines

Steps for Heron’s Formula Applied to SSS triangles Identify the semi-perimeter: 𝒔= 𝒂+𝒃+𝒄 𝟐 To find area, use the equation, 𝑨= 𝒔 𝒔−𝒂 𝒔−𝒃 𝒔−𝒄 Label correctly 11/17/2018 9:48 AM §6.2: Law of Cosines

Example 6 Find the area of the triangle whose sides have the length of 17 ft., 25 ft., and 26 ft. 11/17/2018 9:48 AM §6.2: Law of Cosines

Example 6 Find the area of the triangle whose sides have the length of 17 ft, 25 ft, and 26 ft. 11/17/2018 9:48 AM §6.2: Law of Cosines

Example 7 The distance “as the crow flies” from Los Angeles to New York is 2451 miles, from New York to Montreal is 331 miles, and from Montreal to Los Angeles is 2427 miles. What is the area of the triangular region having these three cities as vertices? (Ignore the curvature of the earth) 11/17/2018 9:48 AM §6.2: Law of Cosines

Your Turn The distance from Springfield to Shelbyville is 43 miles, Shelbyville to Capital City is 53 miles, and Capital City back to Springfield is 43 miles. Eventually, if you draw a shape, it makes an oblique triangle. Determine the area. 11/17/2018 9:48 AM §6.2: Law of Cosines

Assignment Page 415 5-21 EOO, 23, 31-41 odd, 45-53 odd 11/17/2018 9:48 AM §6.2: Law of Cosines