Question

Nghiệm của phương trình sinx-cos2x=2

Answer 1

cos(2 x) + 2 = sin(x)

Solve for x over the real numbers:

sin(x) - cos(2 x) = 2

Transform sin(x) - cos(2 x) into a polynomial with respect to sin(x) using cos(2 x) = 1 - 2 sin^2(x):

-1 + sin(x) + 2 sin^2(x) = 2

Divide both sides by 2:

-1/2 + sin(x)/2 + sin^2(x) = 1

Add 1/2 to both sides:

sin(x)/2 + sin^2(x) = 3/2

Add 1/16 to both sides:

1/16 + sin(x)/2 + sin^2(x) = 25/16

Write the left hand side as a square:

(sin(x) + 1/4)^2 = 25/16

Take the square root of both sides:

sin(x) + 1/4 = 5/4 or sin(x) + 1/4 = -5/4

Subtract 1/4 from both sides:

sin(x) = 1 or sin(x) + 1/4 = -5/4

Take the inverse sine of both sides:

x = 2 π n + π/2 for n element Z

or sin(x) + 1/4 = -5/4

Subtract 1/4 from both sides:

x = 2 π n + π/2 for n element Z

or sin(x) = -3/2

sin(x) = -3/2 has no solution since for all x element R, -1<=sin(x)<=1 and -3/2<-1:

Answer: |

| x = 2 π n + π/2 for n element Z

x = 1/2 (4 π n + π) n element Z