What is the general solution for sin(2x)=sqrt(2)cos(x),0<= x<= 2pi ?

The general solution for sin(2x)=sqrt(2)cos(x),0<= x<= 2pi is x= pi/2 ,x=(3pi)/2 ,x= pi/4 ,x=(3pi)/4

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sin(2x)=sqrt(2)cos(x),0<= x<= 2pi

sin (2 x) = 2 cos (x), 0 \leq x \leq 2 π

x = \frac{π}{2}, x = \frac{3 π}{2}, x = \frac{π}{4}, x = \frac{3 π}{4}

Solution steps

sin (2 x) = 2 cos (x), 0 \leq x \leq 2 π

Subtract

2 cos (x)

from both sides

sin (2 x) - 2 cos (x) = 0

Rewrite using trig identities

- cos (x) 2 + 2 cos (x) sin (x) = 0

Factor

- cos (x) 2 + 2 cos (x) sin (x) : cos (x) (- 2 + 2 sin (x))

cos (x) (- 2 + 2 sin (x)) = 0

Solving each part separately

cos (x) = 0 or - 2 + 2 sin (x) = 0

cos (x) = 0, 0 \leq x \leq 2 π : x = \frac{π}{2}, x = \frac{3 π}{2}

- 2 + 2 sin (x) = 0, 0 \leq x \leq 2 π : x = \frac{π}{4}, x = \frac{3 π}{4}

Combine all the solutions

x = \frac{π}{2}, x = \frac{3 π}{2}, x = \frac{π}{4}, x = \frac{3 π}{4}

y, x = arccos (y)

tan (x) = \frac{- 1}{1}

4 cos (θ) = 1 - cos (θ)

sin (\frac{x}{3}) = 0

cos (x) = 0.95

What is the general solution for sin(2x)=sqrt(2)cos(x),0<= x<= 2pi ?
The general solution for sin(2x)=sqrt(2)cos(x),0<= x<= 2pi is x= pi/2 ,x=(3pi)/2 ,x= pi/4 ,x=(3pi)/4