What is the general solution for 0=3cos(3θ) ?

The general solution for 0=3cos(3θ) is θ= pi/6+(2pin)/3 ,θ= pi/2+(2pin)/3

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0=3cos(3θ)

Solution

0 = 3 cos (3 θ)

Solution

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

Degrees

θ = 3 0^{\circ} + 12 0^{\circ} n, θ = 9 0^{\circ} + 12 0^{\circ} n

Solution steps

0 = 3 cos (3 θ)

Switch sides

3 cos (3 θ) = 0

Divide both sides by

3

3 cos (3 θ) = 0

Divide both sides by

3

\frac{3 cos ( 3 θ )}{3} = \frac{0}{3}

Simplify

cos (3 θ) = 0

cos (3 θ) = 0

General solutions for

cos (3 θ) = 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}

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

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

Solve

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

3 θ = \frac{π}{2} + 2 πn

Divide both sides by

3

3 θ = \frac{π}{2} + 2 πn

Divide both sides by

3

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

Simplify

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

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

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{π}{6} + \frac{2 πn}{3}

θ = \frac{π}{6} + \frac{2 πn}{3}

θ = \frac{π}{6} + \frac{2 πn}{3}

θ = \frac{π}{6} + \frac{2 πn}{3}

Solve

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

3 θ = \frac{3 π}{2} + 2 πn

Divide both sides by

3

3 θ = \frac{3 π}{2} + 2 πn

Divide both sides by

3

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

Simplify

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

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

\frac{\frac{3 π}{2}}{3} = \frac{π}{2}

\frac{\frac{3 π}{2}}{3}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{3 π}{6}

Cancel the common factor:

3

= \frac{π}{2}

= \frac{π}{2} + \frac{2 πn}{3}

θ = \frac{π}{2} + \frac{2 πn}{3}

θ = \frac{π}{2} + \frac{2 πn}{3}

θ = \frac{π}{2} + \frac{2 πn}{3}

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

Popular Examples

sin(1/2 x)=cos(x)

sin (\frac{1}{2} x) = cos (x)

tan(2x-pi/4)=-1

tan (2 x - \frac{π}{4}) = - 1

2sin^2(A)=5sin(A)-2

2 sin^{2} (A) = 5 sin (A) - 2

cot(θ)= 12/5

cot (θ) = \frac{12}{5}

solvefor θ,cos(θ)=-1/2 solve for

θ, cos (θ) = - \frac{1}{2}

Frequently Asked Questions (FAQ)

What is the general solution for 0=3cos(3θ) ?
The general solution for 0=3cos(3θ) is θ= pi/6+(2pin)/3 ,θ= pi/2+(2pin)/3