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5.3. DAY 48 Solving Trig Equations.notebook March 17, 2015 OH nooooooo! DAY 48 March 16/17, 2015 Class Opener: Find D and R: a. y= 5x2+2x3 b. y= 3x2 c. y= 1 +2 x3 d. 2x+5=10 Nov 142:45 PM Learning Goal 5.1.-5.2. QUIZ Trigonometric Identities. Mar 1311:56 AM 1 5.3. DAY 48 Solving Trig Equations.notebook March 17, 2015 IN YOUR NOTES LESSON 5.3. SOLVING TRIGONOMETRIC EQUATIONS Determine whether the following equations are true or false: 1.] 2sin(y) = sin(2y) 2.] cos2(y) = cos(y2) 3.] 3tan(x2) 5tan(x2) = 2 tan (x2) 4.] cos(3y) + cos(y) = cos (4y) 5.] cos(y) + cos(x) = cos (x+y) 6.] sin2(3x) = (sin (3x))2 Feb 142:54 PM LESSON 5.3. SOLVING TRIGONOMETRIC EQUATIONS 1.] 2sin(y) = sin(2y) False 2.] cos2(y) = cos(y2) False 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 Feb 142:54 PM 2 5.3. DAY 48 Solving Trig Equations.notebook March 17, 2015 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. When solving algebraic equations, we can always CHECK (code word for verify) solutions. Example: Verify that each x value is a solution of the equation given: 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 147:51 PM Verifying solutions for Trig Equation: Example: Verify that each x value is a solution of the equation given: Note: solution to the equation is "zero" of the equation 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 3 5.3. DAY 48 Solving Trig Equations.notebook March 17, 2015 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. When solving algebraic equations, we can always CHECK (code word for verify) solutions. Example 1: Solve 2sinx 1 = 0 for x. Feb 283:14 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. When solving algebraic equations, we can always CHECK (code word for verify) solutions. Example 1: Solve 2sinx 1 = 0 for x. There are an infinite number of solutions to this problem. To solve for x, you must first isolate the sine term. therefore The sine function is positive in quadrants I and II. Therefore, two of the solutions to the problem are and Numerical Check: • Left Side: • Right Side: 0 Feb 283:14 PM 4 5.3. DAY 48 Solving Trig Equations.notebook March 17, 2015 Solve 2sinx 1 = 0 for x. Therefore, two of the solutions to the problem are and These two solutions are solution on the interval [0, 2π], but there are infitinely many solutions id the domain is not restricted to a specific interval. There will be 2 solution in every 2π revolutions. If you rotate from π/6 exactly 2π radians, you will hit your next solution. If you rotate from 5π/6 exactly 2π radians, you will hit your next solution. Therefore, ALL GENERAL SOLUTIONS WILL BE where n is the number of revolutions in positive/negative direction. Feb 283:29 PM Solve for x: General solutions: Feb 162:11 PM 5 5.3. DAY 48 Solving Trig Equations.notebook March 17, 2015 Note: General solutions for tangent repeat every π radians! Feb 162:14 PM Feb 162:19 PM 6 5.3. DAY 48 Solving Trig Equations.notebook March 17, 2015 Feb 162:28 PM General Solutions: Feb 173:26 PM 7 5.3. DAY 48 Solving Trig Equations.notebook March 17, 2015 Rewriting as a Single Trigonometric Function: Feb 168:41 AM Remember to use ± when taking the square root!!! Feb 1711:11 AM 8 5.3. DAY 48 Solving Trig Equations.notebook March 17, 2015 Feb 1410:55 PM Lesson 5.3. 364/ 230 EVEN An Answer Key posted on Mrs. Ghillany's LMSA Web Page Feb 169:38 AM 9