What is the general solution for 2+cos(2x)+3cos(x)=0 ?

The general solution for 2+cos(2x)+3cos(x)=0 is x=(2pi)/3+2pin,x=(4pi)/3+2pin,x=pi+2pin

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2+cos(2x)+3cos(x)=0

2 + cos (2 x) + 3 cos (x) = 0

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

Solution steps

2 + cos (2 x) + 3 cos (x) = 0

Rewrite using trig identities

1 + 2 cos^{2} (x) + 3 cos (x) = 0

Solve by substitution

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - 1 : x = π + 2 πn

Combine all the solutions

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

2 cos (x) = tan (x) + sec (x)

sin^{2} (x) cos (x) = \frac{2}{3 π}

tan (φ) = - \frac{1}{6}

sin (θ) = \frac{4}{5} cos (θ)

19 = \frac{1}{2} \cdot 7.9 \cdot 6.2 sin (x)

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