What is the general solution for 2cos(12x)=sqrt(3) ?

The general solution for 2cos(12x)=sqrt(3) is x= pi/(72)+(pin)/6 ,x=(11pi}{72}+\frac{pin)/6

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2cos(12x)=sqrt(3)

Solution

2 cos (12 x) = 3

Solution

x = \frac{π}{72} + \frac{πn}{6}, x = \frac{11 π}{72} + \frac{πn}{6}

Degrees

x = 2. 5^{\circ} + 3 0^{\circ} n, x = 27. 5^{\circ} + 3 0^{\circ} n

Solution steps

2 cos (12 x) = 3

Divide both sides by

2

2 cos (12 x) = 3

Divide both sides by

2

\frac{2 cos ( 12 x )}{2} = \frac{3}{2}

Simplify

cos (12 x) = \frac{3}{2}

cos (12 x) = \frac{3}{2}

General solutions for

cos (12 x) = \frac{3}{2}

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}

12 x = \frac{π}{6} + 2 πn, 12 x = \frac{11 π}{6} + 2 πn

12 x = \frac{π}{6} + 2 πn, 12 x = \frac{11 π}{6} + 2 πn

Solve

12 x = \frac{π}{6} + 2 πn : x = \frac{π}{72} + \frac{πn}{6}

12 x = \frac{π}{6} + 2 πn

Divide both sides by

12

12 x = \frac{π}{6} + 2 πn

Divide both sides by

12

\frac{12 x}{12} = \frac{\frac{π}{6}}{12} + \frac{2 πn}{12}

Simplify

\frac{12 x}{12} = \frac{\frac{π}{6}}{12} + \frac{2 πn}{12}

Simplify

\frac{12 x}{12} : x

\frac{12 x}{12}

Divide the numbers:

\frac{12}{12} = 1

= x

Simplify

\frac{\frac{π}{6}}{12} + \frac{2 πn}{12} : \frac{π}{72} + \frac{πn}{6}

\frac{\frac{π}{6}}{12} + \frac{2 πn}{12}

\frac{\frac{π}{6}}{12} = \frac{π}{72}

\frac{\frac{π}{6}}{12}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 12}

Multiply the numbers:

6 \cdot 12 = 72

= \frac{π}{72}

\frac{2 πn}{12} = \frac{πn}{6}

\frac{2 πn}{12}

Cancel the common factor:

2

= \frac{πn}{6}

= \frac{π}{72} + \frac{πn}{6}

x = \frac{π}{72} + \frac{πn}{6}

x = \frac{π}{72} + \frac{πn}{6}

x = \frac{π}{72} + \frac{πn}{6}

Solve

12 x = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{72} + \frac{πn}{6}

12 x = \frac{11 π}{6} + 2 πn

Divide both sides by

12

12 x = \frac{11 π}{6} + 2 πn

Divide both sides by

12

\frac{12 x}{12} = \frac{\frac{11 π}{6}}{12} + \frac{2 πn}{12}

Simplify

\frac{12 x}{12} = \frac{\frac{11 π}{6}}{12} + \frac{2 πn}{12}

Simplify

\frac{12 x}{12} : x

\frac{12 x}{12}

Divide the numbers:

\frac{12}{12} = 1

= x

Simplify

\frac{\frac{11 π}{6}}{12} + \frac{2 πn}{12} : \frac{11 π}{72} + \frac{πn}{6}

\frac{\frac{11 π}{6}}{12} + \frac{2 πn}{12}

\frac{\frac{11 π}{6}}{12} = \frac{11 π}{72}

\frac{\frac{11 π}{6}}{12}

Apply the fraction rule:

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

= \frac{11 π}{6 \cdot 12}

Multiply the numbers:

6 \cdot 12 = 72

= \frac{11 π}{72}

\frac{2 πn}{12} = \frac{πn}{6}

\frac{2 πn}{12}

Cancel the common factor:

2

= \frac{πn}{6}

= \frac{11 π}{72} + \frac{πn}{6}

x = \frac{11 π}{72} + \frac{πn}{6}

x = \frac{11 π}{72} + \frac{πn}{6}

x = \frac{11 π}{72} + \frac{πn}{6}

x = \frac{π}{72} + \frac{πn}{6}, x = \frac{11 π}{72} + \frac{πn}{6}

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos(12x)=sqrt(3) ?
The general solution for 2cos(12x)=sqrt(3) is x= pi/(72)+(pin)/6 ,x=(11pi}{72}+\frac{pin)/6