What is the general solution for sqrt(3)sin(x)-3cos(x)=sqrt(6) ?

The general solution for sqrt(3)sin(x)-3cos(x)=sqrt(6) is x=-2.87979…+2pin,x=1.83259…+2pin

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sqrt(3)sin(x)-3cos(x)=sqrt(6)

3 sin (x) - 3 cos (x) = 6

x = - 2.87979 \dots + 2 πn, x = 1.83259 \dots + 2 πn

Solution steps

3 sin (x) - 3 cos (x) = 6

Add

3 cos (x)

to both sides

3 sin (x) = 6 + 3 cos (x)

Square both sides

(3 sin (x))^{2} = (6 + 3 cos (x))^{2}

Subtract

(6 + 3 cos (x))^{2}

from both sides

3 sin^{2} (x) - 6 - 66 cos (x) - 9 cos^{2} (x) = 0

Rewrite using trig identities

- 3 - 12 cos^{2} (x) - 6 cos (x) 6 = 0

Solve by substitution

cos (x) = - \frac{6 + 2}{4}, cos (x) = - \frac{6 - 2}{4}

cos (x) = - \frac{6 + 2}{4} : x = arccos (- \frac{6 + 2}{4}) + 2 πn, x = - arccos (- \frac{6 + 2}{4}) + 2 πn

cos (x) = - \frac{6 - 2}{4} : x = arccos (- \frac{6 - 2}{4}) + 2 πn, x = - arccos (- \frac{6 - 2}{4}) + 2 πn

Combine all the solutions

x = arccos (- \frac{6 + 2}{4}) + 2 πn, x = - arccos (- \frac{6 + 2}{4}) + 2 πn, x = arccos (- \frac{6 - 2}{4}) + 2 πn, x = - arccos (- \frac{6 - 2}{4}) + 2 πn

Verify solutions by plugging them into the original equation

x = - arccos (- \frac{6 + 2}{4}) + 2 πn, x = arccos (- \frac{6 - 2}{4}) + 2 πn

Show solutions in decimal form

x = - 2.87979 \dots + 2 πn, x = 1.83259 \dots + 2 πn

tan (θ) = \frac{11}{100}

sin (x) = \frac{4}{41}

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

sin (x) = - 1.1

sin (x) = (\frac{3}{2})

What is the general solution for sqrt(3)sin(x)-3cos(x)=sqrt(6) ?
The general solution for sqrt(3)sin(x)-3cos(x)=sqrt(6) is x=-2.87979…+2pin,x=1.83259…+2pin