What is the general solution for sqrt(6)=-sqrt(8)cos(3x) ?

The general solution for sqrt(6)=-sqrt(8)cos(3x) is x=(5pi}{18}+(2pin)/3 ,x=(7pi)/(18)+\frac{2pin)/3

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sqrt(6)=-sqrt(8)cos(3x)

Solution

6 = - 8 cos (3 x)

Solution

x = \frac{5 π}{18} + \frac{2 πn}{3}, x = \frac{7 π}{18} + \frac{2 πn}{3}

Degrees

x = 5 0^{\circ} + 12 0^{\circ} n, x = 7 0^{\circ} + 12 0^{\circ} n

Solution steps

6 = - 8 cos (3 x)

Switch sides

- 8 cos (3 x) = 6

Simplify

- 8 : - 22

- 8

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

n ab = n a n b

= 2 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 22

= - 22

(- 22) cos (3 x) = 6

Divide both sides by

- 22

(- 22) cos (3 x) = 6

Divide both sides by

- 22

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

Simplify

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

Simplify

\frac{( - 2 2 ) cos ( 3 x )}{- 2 2} : cos (3 x)

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

Remove parentheses:

(- a) = - a

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

Apply the fraction rule:

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

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

2

= cos (3 x)

Simplify

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

\frac{6}{- 2 2}

Apply the fraction rule:

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

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

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

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

General solutions for

cos (3 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}

3 x = \frac{5 π}{6} + 2 πn, 3 x = \frac{7 π}{6} + 2 πn

3 x = \frac{5 π}{6} + 2 πn, 3 x = \frac{7 π}{6} + 2 πn

Solve

3 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{18} + \frac{2 πn}{3}

3 x = \frac{5 π}{6} + 2 πn

Divide both sides by

3

3 x = \frac{5 π}{6} + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3} : \frac{5 π}{18} + \frac{2 πn}{3}

\frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3}

\frac{\frac{5 π}{6}}{3} = \frac{5 π}{18}

\frac{\frac{5 π}{6}}{3}

Apply the fraction rule:

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

= \frac{5 π}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{5 π}{18}

= \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

Solve

3 x = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{18} + \frac{2 πn}{3}

3 x = \frac{7 π}{6} + 2 πn

Divide both sides by

3

3 x = \frac{7 π}{6} + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{7 π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{7 π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{7 π}{6}}{3} + \frac{2 πn}{3} : \frac{7 π}{18} + \frac{2 πn}{3}

\frac{\frac{7 π}{6}}{3} + \frac{2 πn}{3}

\frac{\frac{7 π}{6}}{3} = \frac{7 π}{18}

\frac{\frac{7 π}{6}}{3}

Apply the fraction rule:

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

= \frac{7 π}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{7 π}{18}

= \frac{7 π}{18} + \frac{2 πn}{3}

x = \frac{7 π}{18} + \frac{2 πn}{3}

x = \frac{7 π}{18} + \frac{2 πn}{3}

x = \frac{7 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}, x = \frac{7 π}{18} + \frac{2 πn}{3}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(6)=-sqrt(8)cos(3x) ?
The general solution for sqrt(6)=-sqrt(8)cos(3x) is x=(5pi}{18}+(2pin)/3 ,x=(7pi)/(18)+\frac{2pin)/3