What is the general solution for 14+7cos(2x)=21cos(x) ?

The general solution for 14+7cos(2x)=21cos(x) is x=2pin,x= pi/3+2pin,x=(5pi)/3+2pin

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14+7cos(2x)=21cos(x)

14 + 7 cos (2 x) = 21 cos (x)

x = 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Solution steps

14 + 7 cos (2 x) = 21 cos (x)

Subtract

21 cos (x)

from both sides

14 + 7 cos (2 x) - 21 cos (x) = 0

Rewrite using trig identities

7 + 14 cos^{2} (x) - 21 cos (x) = 0

Solve by substitution

cos (x) = 1, cos (x) = \frac{1}{2}

cos (x) = 1 : x = 2 πn

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

- 8 sin^{2} (θ) - 8 sin (θ) + 12 = - 5 sin (θ) + 7

cos (x) = 1 - 2 cos^{2} (x)

sin (3 x) = 0, 0 \leq x < 2 π

2 cos^{2} (x) = 3 cos (x)

sin (θ) = - \frac{1}{5}, π < θ < \frac{3 π}{2}, cos (θ)

What is the general solution for 14+7cos(2x)=21cos(x) ?
The general solution for 14+7cos(2x)=21cos(x) is x=2pin,x= pi/3+2pin,x=(5pi)/3+2pin