Find the solution in 0 to 2pi of this equation??

Question

Sin 2x = 1 + cos 2x

Answer 1

sin(2x) = 1 + cos(2x)

 

Using double angle identities, the given equation can be rewritten as:

 

2sinxcosx = 1 + (1 - 2sin²x)

 

sinxcosx = 1 - sin²x

 

sinxcosx = cos²x

 

sinxcosx - cos²x = 0

 

cosx(sinx - cosx) = 0

 

cosx = 0 or sinx = cosx

 

cosx = 0 or tanx = 1

 

cosx = 0 when x = π/2, 3π/2

 

tanx = 1 when x = π/4. 5π/4

 

Solutions in [0, 2π]:  π/2, 3π/2, π/4, 5π/4

 

Answer 2

Using your trig identities,

 

2sinxcosx = 1 + (1 - 2sin²x)

 

2sinxcosx = 2 - 2sin²x

 

2sinxcosx = 2(1 - sin²x)

 

2sinxcosx = 2cos²x

 

sinxcosx = cos²x

 

 

Divide both sides of the equation by cos²x.

 

sinx / cosx = 1

 

tan(x) = 1

 

x = π/4

 

x = π + (π/4)

 

x = 5π/4