What is the general solution for cos(2x)=2sin(x) ?

The general solution for cos(2x)=2sin(x) is x=0.37473…+2pin,x=pi-0.37473…+2pin

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cos(2x)=2sin(x)

cos (2 x) = 2 sin (x)

x = 0.37473 \dots + 2 πn, x = π - 0.37473 \dots + 2 πn

Solution steps

cos (2 x) = 2 sin (x)

Subtract

2 sin (x)

from both sides

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

Rewrite using trig identities

1 - 2 sin (x) - 2 sin^{2} (x) = 0

Solve by substitution

sin (x) = - \frac{1 + 3}{2}, sin (x) = \frac{3 - 1}{2}

sin (x) = - \frac{1 + 3}{2} :

No Solution

sin (x) = \frac{3 - 1}{2} : x = arcsin (\frac{3 - 1}{2}) + 2 πn, x = π - arcsin (\frac{3 - 1}{2}) + 2 πn

Combine all the solutions

x = arcsin (\frac{3 - 1}{2}) + 2 πn, x = π - arcsin (\frac{3 - 1}{2}) + 2 πn

Show solutions in decimal form

x = 0.37473 \dots + 2 πn, x = π - 0.37473 \dots + 2 πn

sin (θ) tan (θ) - 3 sin (θ) = 0

53 tan (x) - 5 = 0

x, y = arctan (x + 1)

2 cos (2 θ) + 4 cos (θ) + 2 = - 1

2 cos (2 x + \frac{3 π}{2}) = 3

What is the general solution for cos(2x)=2sin(x) ?
The general solution for cos(2x)=2sin(x) is x=0.37473…+2pin,x=pi-0.37473…+2pin