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
SOLVING TRIGONOMETRIC EQAUTIONS • Solve sin(x) + 2 = 3 for 0° < x < 360° Isolate the variable-containing term: sin(x) + 2 = 3 sin(x) = 1 x = 90° • Solve tan 2(x) + 3 = 0 for 0° < x < 360° tan2(x) = –3 no solution • Solve on 0° < x < 360° To solve this, I need to factor. x = 30°, 90°, 270°, 330° • Copyright © 2010-2011 All Rights Reserved 2(x) – sin(x) = 2 on 0° < x < 360° Solve sin This is a quadratic in sine, so I can factor: sin2(x) – sin(x) – 2 = 0 (sin(x) – 2)(sin(x) + 1) = 0 sin(x) = 2 (not possible!) or sin(x) = –1 Only one of the factor solutions is sensible. For sin(x) = –1, I get: x = 270° • Solve cos 2(x) + cos(x) = sin2(x) on 0° < x < 360° We have two variables so I need use a trig identity to get a quadratic in cosine: 2 2 cos (x) + cos(x) = sin (x) 2 2 cos (x) + cos(x) = 1 – cos (x) 2cos2(x) + cos(x) – 1 = 0 (2cos(x) – 1)(cos(x) + 1) = 0 cos(x) = 1/2 or cos(x) = –1 The first trig equation, cos(x) = 1/2, gives me x = me x = 180°. So my complete solution is: 60° and x = 300°. The second equation gives x = 60°, 180°, 300° • Solve sin(x) = sin(2x) on 0° < x < 360° sin(x) = 2sin(x)cos(x) sin(x) – 2sin(x)cos(x) = 0 Factor sin( x) sin(x)(1 – 2cos(x)) = 0 sin(x) = 0 or 1-2cos(x)=0 cos(x) = 1/2 x = 0°, 60°, 180°, 300°, 360° • Solve sin(x) + cos(x) = 1 on 0° < x < 360° Square both sides. 2 2 (sin(x) + cos(x)) = (1) 2 2 sin (x) + 2sin(x)cos(x) + cos (x) = 1 2 2 [sin (x) + cos (x)] + 2sin(x)cos(x) = 1 1 + 2sin(x)cos(x) = 1 2sin(x)cos(x) = 0 sin(x)cos(x) = 0 x = 0°, 90°, 180°, 270° Plugging back in. sin(0°) + cos(0°) = 0 + 1 = 1 (this solution works) sin(90°) + cos(90°) = 1 + 0 = 1 (this one works, too) sin(180°) + cos(180°) = 0 + (–1) = –1 sin(270°) + cos(270°) = (–1) + 0 = –1 (oh;okay, so this one does NOT work) (this one doesn't work, either) So the actual solution is: x = 0°, 90° EXIT CARD. WHAT DID I LEARN TODAY? ASSESSMENT FOR YOUR LEARNING ____ a. b. 1. ____ a. b. 2. ____ a. b. 3. Which value for is a solution to ? c. d. What quadrants do the solutions to lie in? I, IV c. I, III d. III, IV I, II How many solutions does the equation have for ? 3 c. 4 d. 5 6 ____ 4. places. a. b. Determine the related acute angle to the solutions of accurate to two decimal ____ Use the following graph of to estimate the solution of for . a. b. 5. c. d. 1.57 3.14 c. d. No solution 4.71 ____ 6. Use the following graph of to estimate the solution(s) of for . a. b. 1.57, 4.71 3.14 c. d. 0, 6.28 -1.57 ____ a. b. 7. Factor the expression . ____ a. b. 8. Which of the following is NOT a solution to the equation for ? c. d. ____ a. b. 9. Which of the following is a solution for the equation ? c. d. ____ a. b. 10. Which is a solution for the equation ? c. d. ____ a. b. 11. Solve where . ,, ,, ____ a. b. 12. c. d. c. d. Which is a solution to the equation ? c. d. ASSESSMENT FOR YOUR LEARNING Answer Section MULTIPLE CHOICE no solution ,, ,, 1. ANS: A 2. ANS: C 3. ANS: B 4. ANS: A 5. 6. ANS: D ANS: A 7. ANS: C 8. 9. ANS: D ANS: A 10. ANS: A 11. ANS: A 12. ANS: B