What is the general solution for 0=cos^2(pi/(12)t) ?

The general solution for 0=cos^2(pi/(12)t) is t=6+24n,t=18+24n

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0=cos^2(pi/(12)t)

Solution

0 = cos^{2} (\frac{π}{12} t)

Solution

t = 6 + 24 n, t = 18 + 24 n

Degrees

t = 343.77467 \dots^{\circ} + 1375.09870 \dots^{\circ} n, t = 1031.32403 \dots^{\circ} + 1375.09870 \dots^{\circ} n

Solution steps

0 = cos^{2} (\frac{π}{12} t)

Switch sides

cos^{2} (\frac{π}{12} t) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

cos (\frac{π}{12} t) = 0

General solutions for

cos (\frac{π}{12} t) = 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}

\frac{π}{12} t = \frac{π}{2} + 2 πn, \frac{π}{12} t = \frac{3 π}{2} + 2 πn

\frac{π}{12} t = \frac{π}{2} + 2 πn, \frac{π}{12} t = \frac{3 π}{2} + 2 πn

Solve

\frac{π}{12} t = \frac{π}{2} + 2 πn : t = 6 + 24 n

\frac{π}{12} t = \frac{π}{2} + 2 πn

Multiply both sides by

12

\frac{π}{12} t = \frac{π}{2} + 2 πn

Multiply both sides by

12

12 \cdot \frac{π}{12} t = 12 \cdot \frac{π}{2} + 12 \cdot 2 πn

Simplify

12 \cdot \frac{π}{12} t = 12 \cdot \frac{π}{2} + 12 \cdot 2 πn

Simplify

12 \cdot \frac{π}{12} t : π t

12 \cdot \frac{π}{12} t

Multiply fractions:

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

= \frac{12 π}{12} t

Cancel the common factor:

12

= t π

Simplify

12 \cdot \frac{π}{2} + 12 \cdot 2 πn : 6 π + 24 πn

12 \cdot \frac{π}{2} + 12 \cdot 2 πn

12 \cdot \frac{π}{2} = 6 π

12 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 12}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6 π

12 \cdot 2 πn = 24 πn

12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 24 πn

= 6 π + 24 πn

π t = 6 π + 24 πn

π t = 6 π + 24 πn

π t = 6 π + 24 πn

Divide both sides by

π

π t = 6 π + 24 πn

Divide both sides by

π

\frac{π t}{π} = \frac{6 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} = \frac{6 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{6 π}{π} + \frac{24 πn}{π} : 6 + 24 n

\frac{6 π}{π} + \frac{24 πn}{π}

Cancel

\frac{6 π}{π} : 6

\frac{6 π}{π}

Cancel the common factor:

π

= 6

= 6 + \frac{24 πn}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= 6 + 24 n

t = 6 + 24 n

t = 6 + 24 n

t = 6 + 24 n

Solve

\frac{π}{12} t = \frac{3 π}{2} + 2 πn : t = 18 + 24 n

\frac{π}{12} t = \frac{3 π}{2} + 2 πn

Multiply both sides by

12

\frac{π}{12} t = \frac{3 π}{2} + 2 πn

Multiply both sides by

12

12 \cdot \frac{π}{12} t = 12 \cdot \frac{3 π}{2} + 12 \cdot 2 πn

Simplify

12 \cdot \frac{π}{12} t = 12 \cdot \frac{3 π}{2} + 12 \cdot 2 πn

Simplify

12 \cdot \frac{π}{12} t : π t

12 \cdot \frac{π}{12} t

Multiply fractions:

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

= \frac{12 π}{12} t

Cancel the common factor:

12

= t π

Simplify

12 \cdot \frac{3 π}{2} + 12 \cdot 2 πn : 18 π + 24 πn

12 \cdot \frac{3 π}{2} + 12 \cdot 2 πn

12 \cdot \frac{3 π}{2} = 18 π

12 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 12}{2}

Multiply the numbers:

3 \cdot 12 = 36

= \frac{36 π}{2}

Divide the numbers:

\frac{36}{2} = 18

= 18 π

12 \cdot 2 πn = 24 πn

12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 24 πn

= 18 π + 24 πn

π t = 18 π + 24 πn

π t = 18 π + 24 πn

π t = 18 π + 24 πn

Divide both sides by

π

π t = 18 π + 24 πn

Divide both sides by

π

\frac{π t}{π} = \frac{18 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} = \frac{18 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{18 π}{π} + \frac{24 πn}{π} : 18 + 24 n

\frac{18 π}{π} + \frac{24 πn}{π}

Cancel

\frac{18 π}{π} : 18

\frac{18 π}{π}

Cancel the common factor:

π

= 18

= 18 + \frac{24 πn}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= 18 + 24 n

t = 18 + 24 n

t = 18 + 24 n

t = 18 + 24 n

t = 6 + 24 n, t = 18 + 24 n

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Frequently Asked Questions (FAQ)

What is the general solution for 0=cos^2(pi/(12)t) ?
The general solution for 0=cos^2(pi/(12)t) is t=6+24n,t=18+24n