What is the general solution for 2cos(3t)-1=0 ?

The general solution for 2cos(3t)-1=0 is t= pi/9+(2pin)/3 ,t=(5pi)/9+(2pin)/3

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2cos(3t)-1=0

Solution

2 cos (3 t) - 1 = 0

Solution

t = \frac{π}{9} + \frac{2 πn}{3}, t = \frac{5 π}{9} + \frac{2 πn}{3}

Degrees

t = 2 0^{\circ} + 12 0^{\circ} n, t = 10 0^{\circ} + 12 0^{\circ} n

Solution steps

2 cos (3 t) - 1 = 0

Move

1

to the right side

2 cos (3 t) - 1 = 0

Add

1

to both sides

2 cos (3 t) - 1 + 1 = 0 + 1

Simplify

2 cos (3 t) = 1

2 cos (3 t) = 1

Divide both sides by

2

2 cos (3 t) = 1

Divide both sides by

2

\frac{2 cos ( 3 t )}{2} = \frac{1}{2}

Simplify

cos (3 t) = \frac{1}{2}

cos (3 t) = \frac{1}{2}

General solutions for

cos (3 t) = \frac{1}{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 t = \frac{π}{3} + 2 πn, 3 t = \frac{5 π}{3} + 2 πn

3 t = \frac{π}{3} + 2 πn, 3 t = \frac{5 π}{3} + 2 πn

Solve

3 t = \frac{π}{3} + 2 πn : t = \frac{π}{9} + \frac{2 πn}{3}

3 t = \frac{π}{3} + 2 πn

Divide both sides by

3

3 t = \frac{π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 t}{3} : t

\frac{3 t}{3}

Divide the numbers:

\frac{3}{3} = 1

= t

Simplify

\frac{\frac{π}{3}}{3} + \frac{2 πn}{3} : \frac{π}{9} + \frac{2 πn}{3}

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

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= \frac{π}{9} + \frac{2 πn}{3}

t = \frac{π}{9} + \frac{2 πn}{3}

t = \frac{π}{9} + \frac{2 πn}{3}

t = \frac{π}{9} + \frac{2 πn}{3}

Solve

3 t = \frac{5 π}{3} + 2 πn : t = \frac{5 π}{9} + \frac{2 πn}{3}

3 t = \frac{5 π}{3} + 2 πn

Divide both sides by

3

3 t = \frac{5 π}{3} + 2 πn

Divide both sides by

3

\frac{3 t}{3} = \frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 t}{3} = \frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 t}{3} : t

\frac{3 t}{3}

Divide the numbers:

\frac{3}{3} = 1

= t

Simplify

\frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3} : \frac{5 π}{9} + \frac{2 πn}{3}

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

\frac{\frac{5 π}{3}}{3} = \frac{5 π}{9}

\frac{\frac{5 π}{3}}{3}

Apply the fraction rule:

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

= \frac{5 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{5 π}{9}

= \frac{5 π}{9} + \frac{2 πn}{3}

t = \frac{5 π}{9} + \frac{2 πn}{3}

t = \frac{5 π}{9} + \frac{2 πn}{3}

t = \frac{5 π}{9} + \frac{2 πn}{3}

t = \frac{π}{9} + \frac{2 πn}{3}, t = \frac{5 π}{9} + \frac{2 πn}{3}

Popular Examples

sin(x)+cos(x)-1=0

sin (x) + cos (x) - 1 = 0

tan(θ)=-9

tan (θ) = - 9

tan(θ)=13

tan (θ) = 13

tan(θ)=16

tan (θ) = 16

cos(θ)=-1/5

cos (θ) = - \frac{1}{5}

Frequently Asked Questions (FAQ)

What is the general solution for 2cos(3t)-1=0 ?
The general solution for 2cos(3t)-1=0 is t= pi/9+(2pin)/3 ,t=(5pi)/9+(2pin)/3