4cos^3(x)=3cos(x)

Solution

4 cos^{3} (x) = 3 cos (x)

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

4 cos^{3} (x) = 3 cos (x)

Solve by substitution

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

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

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

cos (x) = \frac{3}{2} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

7 cos (2 t) = 3

sinh^{2} (x) - 5 cosh (x) = - 7

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

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

cos^{2} (x) = \frac{( 8 k - 9 )}{2}