Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
5.3. Solving Trig Equations.notebook February 17, 2012 The Ferris Wheel Story LESSON 5.3. SOLVING TRIGONOMETRIC EQUATIONS True or false: False False 1.] 2sin(y) = sin(2y) 2.] cos2(y) = cos(y2) 3.] 3tan(x2) 5tan(x2) = 2 tan (x2) True 4.] cos(3y) + cos(y) = cos (4y) False 5.] cos(y) + cos(x) = cos (x+y) False 6.] sin2(3x) = (sin (3x))2 True The 1893 Chicago World’s Fair is considered the birthplace of the classic amusement park ride, the Ferris wheel. The architectural wonder was created by an American engineer named George Ferris. The original Ferris wheel no longer exists. But, in 1990, a new Ferris wheel was built at Navy Pier in Chicago to resemble the original. While the Navy Pier Ferris Wheel is a beautiful Chicago landmark, its grandeur actually pales in comparison to Mr. Ferris’ creation. The Ferris wheel built for the World’s Fair had a diameter of 250 feet. It stood 14 feet off the ground. It had 36 wooden boxcars that were the size of train cars. Each car could hold 60 people! The wheel would load cars in such a way that each rider could enjoy a full rotation that lasted about 6 minutes. The Ferris wheel at Navy Pier has a diameter of 140 feet. It stands 30 feet off the ground. The wheel has 40 gondolas that seat six passengers each. It takes about 6 minutes for the Navy Pier Ferris Wheel to complete one rotation. Above is a picture of the first Ferris wheel next to the Ferris wheel at Navy Pier. Feb 142:54 PM Feb 142:57 PM Using the given information, the equations representing the two Ferris Wheels are as follows World's Fair Ferris Wheel Navy Pier Ferris Wheel Imagine the Navy Pier and the World’s Fair Ferris Wheel being built beside each other. If you got on the Navy Pier Ferris Wheel at the same time that your best friend got on the World's Fair Ferris Wheel, when would you be at the same height? If the wheels are turning at the same speeds, at what times over a 15 minute time period would you be at the same height? However, due to a cyclical pattern of trigonometric functions there will be infinitely many solutions. Feb 142:58 PM Feb 148:14 PM Lesson 5.3. Solving Trigonometric Equations (Easy cases) When solving a trigonometric equation, your goal is to isolate the trigonometric function involved in the equation using standard algebraic operations and trigonometric identities. So, how many times will the two Ferris Wheels be at the same height over the course of 15 minutes? Feb 156:11 AM When solving algebraic equations, we can always CHECK (code word for verify) solutions. Feb 147:51 PM 1 5.3. Solving Trig Equations.notebook Verifying solutions for Trig Equation: Example: Verify that each x value is a solution of the equation given: February 17, 2012 SOLVING EQUATIONS. Find solutions for the following trigonometric equations on interval and ALL general solutions. 1.] 2 cos(x) – 1 = 0 (a) x = π/3 (b) x = 5π/3 2.] sec x – 2 = 0 (a) x = π/3 (b) x = 5π/3 Feb 1410:43 PM Feb 1410:44 PM Feb 162:11 PM Feb 162:14 PM Equations of Quadratic type: Feb 162:19 PM Feb 1410:49 PM 2 5.3. Solving Trig Equations.notebook Feb 173:17 PM February 17, 2012 Feb 162:28 PM Extracting Square Roots: Remember to use ± when taking the square root!!! Feb 168:32 AM Feb 173:26 PM Rewriting as a Single Trigonometric Function: Feb 168:41 AM Feb 1712:09 PM 3 5.3. Solving Trig Equations.notebook February 17, 2012 Functions of Multiple Angles: Remember to use ± when taking the square root!!! Feb 1711:11 AM Feb 168:42 AM Try these in your groups: Page 364/ 1, 3, 5 Page 364/ 7, 13, 17, 23, 29 . OTL Page 364/2230 EVEN Page 366/ 68, 72 TAKE HOME TEST Feb 1711:19 AM Feb 1410:55 PM Feb 169:38 AM 4