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

Solution

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

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Solution steps

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

Rewrite using trig identities

- cos (x) + 4 cos^{3} (x) = 0

Solve by substitution

cos (x) = 0, cos (x) = - \frac{1}{2}, cos (x) = \frac{1}{2}

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

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

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

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

tan (x) = \frac{\frac{3}{2}}{\frac{1}{2}}

tan (2 x) + sec (2 x) = 11

- 8 sin (x) + 8 cos (2 x) = 0

tan^{2} (x) + 2 sec (x) + 2 = 0