What is the general solution for (sqrt(6))/3 cos(x)= 1/(sqrt(2)) ?

The general solution for (sqrt(6))/3 cos(x)= 1/(sqrt(2)) is x= pi/6+2pin,x=(11pi)/6+2pin

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(sqrt(6))/3 cos(x)= 1/(sqrt(2))

Solution

\frac{6}{3} cos (x) = \frac{1}{2}

Solution

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

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{6}{3} cos (x) = \frac{1}{2}

Multiply both sides by

3

\frac{6}{3} cos (x) = \frac{1}{2}

Multiply both sides by

3

3 \cdot \frac{6}{3} cos (x) = \frac{1 \cdot 3}{2}

Simplify

3 \cdot \frac{6}{3} cos (x) = \frac{1 \cdot 3}{2}

Simplify

3 \cdot \frac{6}{3} cos (x) : 6 cos (x)

3 \cdot \frac{6}{3} cos (x)

Multiply fractions:

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

= \frac{6 \cdot 3 cos ( x )}{3}

Cancel the common factor:

3

= 6 cos (x)

Simplify

\frac{1 \cdot 3}{2} : \frac{3 2}{2}

\frac{1 \cdot 3}{2}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{2}

Rationalize

\frac{3}{2} : \frac{3 2}{2}

\frac{3}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{3 2}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{3 2}{2}

= \frac{3 2}{2}

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

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

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

Divide both sides by

6

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

Divide both sides by

6

\frac{6 cos ( x )}{6} = \frac{\frac{3 2}{2}}{6}

Simplify

\frac{6 cos ( x )}{6} = \frac{\frac{3 2}{2}}{6}

Simplify

\frac{6 cos ( x )}{6} : cos (x)

\frac{6 cos ( x )}{6}

Cancel the common factor:

6

= cos (x)

Simplify

\frac{\frac{3 2}{2}}{6} : \frac{3}{2}

\frac{\frac{3 2}{2}}{6}

Apply the fraction rule:

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

= \frac{3 2}{2 6}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

n ab = n a n b

= 23

= \frac{3 2}{2 2 3}

Cancel

\frac{3 2}{2 2 3} : \frac{3 2}{2 2}

\frac{3 2}{2 2 3}

Apply radical rule:

n a = a^{\frac{1}{n}}

3 = 3^{\frac{1}{2}}

= \frac{3 2}{2 2 \cdot 3 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= \frac{2 \cdot 3 ^{- \frac{1}{2} + 1}}{2 2}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{3 ^{\frac{1}{2}} 2}{2 2}

Apply radical rule:

a^{\frac{1}{n}} = n a

3^{\frac{1}{2}} = 3

= \frac{3 2}{2 2}

= \frac{3 2}{2 2}

Simplify

32 : 6

32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= \frac{6}{2 2}

Rationalize

\frac{6}{2 2} : \frac{3}{2}

\frac{6}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{6 2}{2 2 2}

62 = 23

62

Factor integer

6 = 2 \cdot 3

= 2 \cdot 3 2

Apply radical rule:

n ab = n a n b

2 \cdot 3 = 23

= 232

Apply radical rule:

a a = a

22 = 2

= 23

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 3}{4}

Cancel the common factor:

2

= \frac{3}{2}

= \frac{3}{2}

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

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

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

General solutions for

cos (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}

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

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for (sqrt(6))/3 cos(x)= 1/(sqrt(2)) ?
The general solution for (sqrt(6))/3 cos(x)= 1/(sqrt(2)) is x= pi/6+2pin,x=(11pi)/6+2pin