What is the general solution for cos(nx)=0 ?

The general solution for cos(nx)=0 is x= pi/(2n)+(2pik)/n ,x=(3pi}{2n}+\frac{2pik)/n ;\quad n\ne 0

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cos(nx)=0

cos (n x) = 0

x = \frac{π}{2 n} + \frac{2 πk}{n}, x = \frac{3 π}{2 n} + \frac{2 πk}{n}; n \neq = 0

Solution steps

cos (n x) = 0

General solutions for

cos (n x) = 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}

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

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

Solve

n x = \frac{π}{2} + 2 πk : x = \frac{π}{2 n} + \frac{2 πk}{n}; n \neq = 0

n x = \frac{π}{2} + 2 πk

Divide both sides by

n; n \neq = 0

n x = \frac{π}{2} + 2 πk

Divide both sides by

n; n \neq = 0

\frac{n x}{n} = \frac{\frac{π}{2}}{n} + \frac{2 πk}{n}; n \neq = 0

Simplify

\frac{n x}{n} = \frac{\frac{π}{2}}{n} + \frac{2 πk}{n}

Simplify

\frac{n x}{n} : x

\frac{n x}{n}

Cancel the common factor:

n

= x

Simplify

\frac{\frac{π}{2}}{n} + \frac{2 πk}{n} : \frac{π}{2 n} + \frac{2 πk}{n}

\frac{\frac{π}{2}}{n} + \frac{2 πk}{n}

Apply the fraction rule:

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

= \frac{π}{2 n} + \frac{2 πk}{n}

x = \frac{π}{2 n} + \frac{2 πk}{n}; n \neq = 0

x = \frac{π}{2 n} + \frac{2 πk}{n}; n \neq = 0

x = \frac{π}{2 n} + \frac{2 πk}{n}; n \neq = 0

Solve

n x = \frac{3 π}{2} + 2 πk : x = \frac{3 π}{2 n} + \frac{2 πk}{n}; n \neq = 0

n x = \frac{3 π}{2} + 2 πk

Divide both sides by

n; n \neq = 0

n x = \frac{3 π}{2} + 2 πk

Divide both sides by

n; n \neq = 0

\frac{n x}{n} = \frac{\frac{3 π}{2}}{n} + \frac{2 πk}{n}; n \neq = 0

Simplify

\frac{n x}{n} = \frac{\frac{3 π}{2}}{n} + \frac{2 πk}{n}

Simplify

\frac{n x}{n} : x

\frac{n x}{n}

Cancel the common factor:

n

= x

Simplify

\frac{\frac{3 π}{2}}{n} + \frac{2 πk}{n} : \frac{3 π}{2 n} + \frac{2 πk}{n}

\frac{\frac{3 π}{2}}{n} + \frac{2 πk}{n}

Apply the fraction rule:

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

= \frac{3 π}{2 n} + \frac{2 πk}{n}

x = \frac{3 π}{2 n} + \frac{2 πk}{n}; n \neq = 0

x = \frac{3 π}{2 n} + \frac{2 πk}{n}; n \neq = 0

x = \frac{3 π}{2 n} + \frac{2 πk}{n}; n \neq = 0

x = \frac{π}{2 n} + \frac{2 πk}{n}, x = \frac{3 π}{2 n} + \frac{2 πk}{n}; n \neq = 0

cos^{3} (θ) = cos (θ)

4 cos (2 x) = 22

4 sin^{4} (x) - 13 sin^{2} (x) + 3 = 0

3 tan (\frac{2 x}{3}) - 1 = 0

8 cos^{2} (θ) - 3 = 1

What is the general solution for cos(nx)=0 ?
The general solution for cos(nx)=0 is x= pi/(2n)+(2pik)/n ,x=(3pi}{2n}+\frac{2pik)/n ;\quad n\ne 0