What is the general solution for cos(x^3)=0 ?

The general solution for cos(x^3)=0 is x=\sqrt[3]{(pi+4pin)/2},x=\sqrt[3]{(3pi+4pin)/2}

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cos(x^3)=0

cos (x^{3}) = 0

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

Degrees

x = 0^{\circ} + 113.89008 \dots^{\circ} n, x = 0^{\circ} + 127.40758 \dots^{\circ} n

Solution steps

cos (x^{3}) = 0

General solutions for

cos (x^{3}) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

Solve

x^{3} = \frac{π}{2} + 2 πn : x = 3 \frac{π + 4 πn}{2}

x^{3} = \frac{π}{2} + 2 πn

For

x^{n} = f (a)

, n is odd, the solution is

x = n f (a)

x = 3 \frac{π}{2} + 2 πn

Simplify

3 \frac{π}{2} + 2 πn : 3 \frac{π + 4 πn}{2}

3 \frac{π}{2} + 2 πn

Join

\frac{π}{2} + 2 πn : \frac{π + 4 πn}{2}

\frac{π}{2} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

= \frac{π}{2} + \frac{2 πn \cdot 2}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + 2 πn \cdot 2}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn}{2}

= 3 \frac{π + 4 πn}{2}

x = 3 \frac{π + 4 πn}{2}