What is the general solution for 4cos(3θ)+2sqrt(3)=0 ?

The general solution for 4cos(3θ)+2sqrt(3)=0 is θ=(5pi}{18}+(2pin)/3 ,θ=(7pi)/(18)+\frac{2pin)/3

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4cos(3θ)+2sqrt(3)=0

Solution

4 cos (3 θ) + 23 = 0

Solution

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

Degrees

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

Solution steps

4 cos (3 θ) + 23 = 0

Move

23

to the right side

4 cos (3 θ) + 23 = 0

Subtract

23

from both sides

4 cos (3 θ) + 23 - 23 = 0 - 23

Simplify

4 cos (3 θ) = - 23

4 cos (3 θ) = - 23

Divide both sides by

4

4 cos (3 θ) = - 23

Divide both sides by

4

\frac{4 cos ( 3 θ )}{4} = \frac{- 2 3}{4}

Simplify

\frac{4 cos ( 3 θ )}{4} = \frac{- 2 3}{4}

Simplify

\frac{4 cos ( 3 θ )}{4} : cos (3 θ)

\frac{4 cos ( 3 θ )}{4}

Divide the numbers:

\frac{4}{4} = 1

= cos (3 θ)

Simplify

\frac{- 2 3}{4} : - \frac{3}{2}

\frac{- 2 3}{4}

Apply the fraction rule:

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

= - \frac{2 3}{4}

Cancel the common factor:

2

= - \frac{3}{2}

cos (3 θ) = - \frac{3}{2}

cos (3 θ) = - \frac{3}{2}

cos (3 θ) = - \frac{3}{2}

General solutions for

cos (3 θ) = - \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 θ = \frac{5 π}{6} + 2 πn, 3 θ = \frac{7 π}{6} + 2 πn

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

Solve

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

3 θ = \frac{5 π}{6} + 2 πn

Divide both sides by

3

3 θ = \frac{5 π}{6} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

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}

θ = \frac{5 π}{18} + \frac{2 πn}{3}

θ = \frac{5 π}{18} + \frac{2 πn}{3}

θ = \frac{5 π}{18} + \frac{2 πn}{3}

Solve

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

3 θ = \frac{7 π}{6} + 2 πn

Divide both sides by

3

3 θ = \frac{7 π}{6} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

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}

θ = \frac{7 π}{18} + \frac{2 πn}{3}

θ = \frac{7 π}{18} + \frac{2 πn}{3}

θ = \frac{7 π}{18} + \frac{2 πn}{3}

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4cos(3θ)+2sqrt(3)=0 ?
The general solution for 4cos(3θ)+2sqrt(3)=0 is θ=(5pi}{18}+(2pin)/3 ,θ=(7pi)/(18)+\frac{2pin)/3