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

The general solution for 3cos^2(x)-sin(x)-1=0 is x=(3pi)/2+2pin,x=0.72972…+2pin,x=pi-0.72972…+2pin

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

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

x = \frac{3 π}{2} + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

Solution steps

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

Rewrite using trig identities

2 - sin (x) - 3 sin^{2} (x) = 0

Solve by substitution

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = \frac{3 π}{2} + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

tan^{2} (θ) sin (θ) = tan^{2} (θ)

cos (x) = \frac{1}{cot ( x )}

(sin (x) - 1) \cdot (sin (x) - 3) = 0

2 - 3 cos (θ) = 0

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

What is the general solution for 3cos^2(x)-sin(x)-1=0 ?
The general solution for 3cos^2(x)-sin(x)-1=0 is x=(3pi)/2+2pin,x=0.72972…+2pin,x=pi-0.72972…+2pin