What is the general solution for 3cos(2x)-2cos(x)-1=0 ?

The general solution for 3cos(2x)-2cos(x)-1=0 is x=2pin,x=2.30052…+2pin,x=-2.30052…+2pin

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3cos(2x)-2cos(x)-1=0

3 cos (2 x) - 2 cos (x) - 1 = 0

x = 2 πn, x = 2.30052 \dots + 2 πn, x = - 2.30052 \dots + 2 πn

Solution steps

3 cos (2 x) - 2 cos (x) - 1 = 0

Rewrite using trig identities

- 4 - 2 cos (x) + 6 cos^{2} (x) = 0

Solve by substitution

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

cos (x) = 1 : x = 2 πn

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

Combine all the solutions

x = 2 πn, x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = 2.30052 \dots + 2 πn, x = - 2.30052 \dots + 2 πn

sin (3 π x) = 0

tan (2 t) = 1, 0 \leq t < 2 π

sin^{2} (x) = cot (x)

sin (\frac{x}{2}) = 1, 0 \leq x < 2 π

sin (\frac{x}{2}) = - cos (x - \frac{π}{3})

What is the general solution for 3cos(2x)-2cos(x)-1=0 ?
The general solution for 3cos(2x)-2cos(x)-1=0 is x=2pin,x=2.30052…+2pin,x=-2.30052…+2pin