What is the general solution for 3sin(2x)-9sin(x)-4cos(x)+6=0 ?

The general solution for 3sin(2x)-9sin(x)-4cos(x)+6=0 is x=0.72972…+2pin,x=pi-0.72972…+2pin

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3sin(2x)-9sin(x)-4cos(x)+6=0

3 sin (2 x) - 9 sin (x) - 4 cos (x) + 6 = 0

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

Solution steps

3 sin (2 x) - 9 sin (x) - 4 cos (x) + 6 = 0

Rewrite using trig identities

6 - 4 cos (x) - 9 sin (x) + 6 cos (x) sin (x) = 0

Factor

6 - 4 cos (x) - 9 sin (x) + 6 cos (x) sin (x) : (2 cos (x) - 3) (3 sin (x) - 2)

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

Solving each part separately

2 cos (x) - 3 = 0 or 3 sin (x) - 2 = 0

2 cos (x) - 3 = 0 :

No Solution

3 sin (x) - 2 = 0 : x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

Combine all the solutions

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

Show solutions in decimal form

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

\frac{10}{sin ( x )} = \frac{13}{sin ( 7 2 ^{\circ} )}

sin (x) - 3 cos (x) = 1, 0 \leq x < 2 π

sin (α) + 1 = cos (α)

\frac{sin ( 11 5 ^{\circ} )}{53} = \frac{sin ( S )}{83}

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

What is the general solution for 3sin(2x)-9sin(x)-4cos(x)+6=0 ?
The general solution for 3sin(2x)-9sin(x)-4cos(x)+6=0 is x=0.72972…+2pin,x=pi-0.72972…+2pin