2cos(3x)=cot(3x)

Solution

2 cos (3 x) = cot (3 x)

x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}, x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

Solution steps

2 cos (3 x) = cot (3 x)

Subtract

cot (3 x)

from both sides

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

Express with sin, cos

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

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

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

Factor

- cos (3 x) + 2 cos (3 x) sin (3 x) : cos (3 x) (2 sin (3 x) - 1)

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

Solving each part separately

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

cos (3 x) = 0 : x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

2 sin (3 x) - 1 = 0 : x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

Combine all the solutions

x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}, x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

cos (x) (cos (x) - 1) = 0, n = 0

1 + cos^{2} (x) = 0

tan (2 x) = - 1.238

tan (θ) = π

csc^{2} (x) + 2 cot^{2} (x) = 1