SOLUTION: cos(4𝑥) − 3 sin (3𝜋/2+ 2𝑥) + 2 = 0 Find the general solutions (in radians)

Algebra ->  Trigonometry-basics -> SOLUTION: cos(4𝑥) − 3 sin (3𝜋/2+ 2𝑥) + 2 = 0 Find the general solutions (in radians)      Log On


   

Question 1204390: cos(4𝑥) − 3 sin (3𝜋/2+ 2𝑥) + 2 = 0
Find the general solutions (in radians)
Found 2 solutions by MathLover1, Edwin McCravy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

cos%284x%29+-+3sin+%283pi%2F2%2B+2x%29+%2B+2+=+0
using trigonometric identities to simplify




1+-+2cos%5E2%28x%29+%2B+2cos%5E4%28x%29%2B+6cos%5E2%28x%29%281+-+sin%5E2%28x%29%29+-1=0}
1+-+2cos%5E2%28x%29+%2B+2+cos%5E4%28x%29%2B+6cos%5E2%28x%29%2Acos%5E2%28x%29+-1=0
+-+2+cos%5E2%28x%29+%2B+2+cos%5E4%28x%29%2B+6cos%5E4%28x%29+=0
8cos%5E4%28x%29-+2+cos%5E2%28x%29++=0
2cos%5E2%28x%29+%284cos%5E2%28x%29-+1%29+=0

solutions:
if 2cos%5E2%28x%29++=0 => x+=pi%2An+%2B+pi%2F2, n element Z
if %284cos%5E2%28x%29-+1%29+=0+=> x+=+pi%2An+-+pi%2F3, n element Z , and x+=+pi%2An+%2Bpi%2F3, n element Z
combine solutions:
x+=pi%2An+%2B+pi%2F2
x+=+pi%2An+-+pi%2F3
x+=+pi%2An+%2Bpi%2F3


Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
cos%284x%29+-+3+sin+%283pi%2F2%2B+2x%29+%2B+2%22%22=%22%220

%22%22=%22%220

2cos%5E2%282x%29-1+-+3%28%28-1%29cos%282x%29%2B%280%29sin%282x%29%5E%22%22%5E%22%22%29+%2B+2%22%22=%22%220

2cos%5E2%282x%29+-+1+%2B+3cos%282x%29+%2B+2%22%22=%22%220

2cos%5E2%282x%29+%2B+3cos%282x%29+%2B+1%22%22=%22%220

%282cos%282x%29%2B1%29%28cos%282x%29+%2B+1%29%22%22=%22%220

2cos(2x)+1 = 0;  cos(2x) + 1 = 0
  2cos(2x) = -1;     cos(2x) = -1
   cos(2x) = -1/2

+cos%282x%29+=+-1%2F2
 

matrix%281%2C5%2C+++x%2C%22%22=%22%22%2C+pi%281%2F3+%2B+n%29%2C+or%2C+pi%282%2F3%2Bn%29%29

+cos%282x%29+=+-1
matrix%281%2C3%2C2x%2C%22%22=%22%22%2C+pi+%2B+2n%2Api%29 
matrix%281%2C3%2Cx%2C%22%22=%22%22%2Cpi%281%2B2n%29%29

Edwin