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How do you find x?

90 = 40.2443sin (0.0729x + 4.6878) + 40.3295

 Apr 13, 2016
 #1
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40.2443sin (0.0729x + 4.6878) + 40.3295 = 90

40.2443sin (0.0729x + 4.6878) = 49.6705

sin (0.0729x + 4.6878) = 49.6705/40.2443 (= 1.234224474024893)

Because the function sine will only give out answers between -1 and 1

Therefore sin (0.0729x + 4.6878) = 1.234224474024893 is impossible

So there are no solutions to this equation

 Apr 13, 2016
 #2
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Solve for x:
90 = 40.3295+40.2443 sin(0.0729 x+4.6878)

40.3295+40.2443 sin(0.0729 x+4.6878) = 80659/2000+(402443 sin((729 x)/10000+23439/5000))/10000:
90 = 80659/2000+(402443 sin((729 x)/10000+23439/5000))/10000

90 = 80659/2000+(402443 sin((729 x)/10000+23439/5000))/10000 is equivalent to 80659/2000+(402443 sin((729 x)/10000+23439/5000))/10000 = 90:
80659/2000+(402443 sin((729 x)/10000+23439/5000))/10000 = 90

Subtract 80659/2000 from both sides:
(402443 sin((729 x)/10000+23439/5000))/10000 = 99341/2000

Multiply both sides by 10000/402443:
sin((729 x)/10000+23439/5000) = 496705/402443

Take the inverse sine of both sides:
(729 x)/10000+23439/5000 = pi-sin^(-1)(496705/402443)+2 pi n_1  for  n_1  element Z
   or  (729 x)/10000+23439/5000 = sin^(-1)(496705/402443)+2 pi n_2  for  n_2  element Z

Subtract 23439/5000 from both sides:
(729 x)/10000 = -23439/5000+pi-sin^(-1)(496705/402443)+2 pi n_1  for  n_1  element Z
   or  (729 x)/10000+23439/5000 = sin^(-1)(496705/402443)+2 pi n_2  for  n_2  element Z

Multiply both sides by 10000/729:
x = -15626/243+(10000 pi)/729-10000/729 sin^(-1)(496705/402443)+(20000 pi n_1)/729  for  n_1  element Z
   or  (729 x)/10000+23439/5000 = sin^(-1)(496705/402443)+2 pi n_2  for  n_2  element Z

Subtract 23439/5000 from both sides:
x = -15626/243+(10000 pi)/729-10000/729 sin^(-1)(496705/402443)+(20000 pi n_1)/729  for  n_1  element Z
   or  (729 x)/10000 = -23439/5000+sin^(-1)(496705/402443)+2 pi n_2  for  n_2  element Z

Multiply both sides by 10000/729:
Answer: |  x = -15626/243+(10000 pi)/729-10000/729 sin^(-1)(496705/402443)+(20000 pi n_1)/729  for  n_1  element Z
   or  x = -15626/243+10000/729 sin^(-1)(496705/402443)+(20000 pi n_2)/729  for  n_2  element Z

 Apr 13, 2016

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