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

The general solution for 6sin(x)=cos(x)-3 is x=-2.46068…+2pin,x=2pi-0.35061…+2pin

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

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

x = - 2.46068 \dots + 2 πn, x = 2 π - 0.35061 \dots + 2 πn

Solution steps

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

Square both sides

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

Subtract

(cos (x) - 3)^{2}

from both sides

36 sin^{2} (x) - cos^{2} (x) + 6 cos (x) - 9 = 0

Rewrite using trig identities

27 - 37 cos^{2} (x) + 6 cos (x) = 0

Solve by substitution

cos (x) = - \frac{3 ( 4 7 - 1 )}{37}, cos (x) = \frac{3 ( 1 + 4 7 )}{37}

cos (x) = - \frac{3 ( 4 7 - 1 )}{37} : x = arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn, x = - arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn

cos (x) = \frac{3 ( 1 + 4 7 )}{37} : x = arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn, x = 2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn

Combine all the solutions

x = arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn, x = - arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn, x = arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn, x = 2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn

Verify solutions by plugging them into the original equation

x = - arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn, x = 2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn

Show solutions in decimal form

x = - 2.46068 \dots + 2 πn, x = 2 π - 0.35061 \dots + 2 πn

- 2 sin (4 x) = 0

2 sin (x) + 3 = 0, 0 \leq x \leq 2 π

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

0 = 1 + 2 sin (θ)

csc (x - \frac{π}{3}) = 2

What is the general solution for 6sin(x)=cos(x)-3 ?
The general solution for 6sin(x)=cos(x)-3 is x=-2.46068…+2pin,x=2pi-0.35061…+2pin