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
