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

The general solution for cot(x)+6sin(x)-2cos(x)=3 is x= pi/6+2pin,x=(5pi)/6+2pin,x=0.32175…+pin

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cot(x)+6sin(x)-2cos(x)=3

cot (x) + 6 sin (x) - 2 cos (x) = 3

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = 0.32175 \dots + πn

Solution steps

cot (x) + 6 sin (x) - 2 cos (x) = 3

Subtract

3

from both sides

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

Express with sin, cos

\frac{cos ( x ) - 3 sin ( x ) + 6 sin ^{2} ( x ) - 2 cos ( x ) sin ( x )}{sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Factor

cos (x) - 3 sin (x) + 6 sin^{2} (x) - 2 cos (x) sin (x) : (1 - 2 sin (x)) (cos (x) - 3 sin (x))

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

Solving each part separately

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

1 - 2 sin (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

cos (x) - 3 sin (x) = 0 : x = arctan (\frac{1}{3}) + πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = arctan (\frac{1}{3}) + πn

Show solutions in decimal form

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = 0.32175 \dots + πn

cos (2 θ) = cos^{2} (θ)

cos (a) = \frac{12}{13}

π cos (π x) = 0

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

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

What is the general solution for cot(x)+6sin(x)-2cos(x)=3 ?
The general solution for cot(x)+6sin(x)-2cos(x)=3 is x= pi/6+2pin,x=(5pi)/6+2pin,x=0.32175…+pin