What is the general solution for cos(θ-pi/6)=(sqrt(3))/2 ?

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

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cos(θ-pi/6)=(sqrt(3))/2

Solution

cos (θ - \frac{π}{6}) = \frac{3}{2}

Solution

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

Degrees

θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 36 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (θ - \frac{π}{6}) = \frac{3}{2}

General solutions for

cos (θ - \frac{π}{6}) = \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}

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

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

Solve

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

θ - \frac{π}{6} = \frac{π}{6} + 2 πn

Move

\frac{π}{6}

to the right side

θ - \frac{π}{6} = \frac{π}{6} + 2 πn

Add

\frac{π}{6}

to both sides

θ - \frac{π}{6} + \frac{π}{6} = \frac{π}{6} + 2 πn + \frac{π}{6}

Simplify

θ - \frac{π}{6} + \frac{π}{6} = \frac{π}{6} + 2 πn + \frac{π}{6}

Simplify

θ - \frac{π}{6} + \frac{π}{6} : θ

θ - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} = 0

= θ

Simplify

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

\frac{π}{6} + 2 πn + \frac{π}{6}

Group like terms

= \frac{π}{6} + \frac{π}{6} + 2 πn

Combine the fractions

\frac{π}{6} + \frac{π}{6} : \frac{π}{3}

Apply rule

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

= \frac{π + π}{6}

Add similar elements:

π + π = 2 π

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

= \frac{π}{3} + 2 πn

θ = \frac{π}{3} + 2 πn

θ = \frac{π}{3} + 2 πn

θ = \frac{π}{3} + 2 πn

Solve

θ - \frac{π}{6} = \frac{11 π}{6} + 2 πn : θ = 2 πn + 2 π

θ - \frac{π}{6} = \frac{11 π}{6} + 2 πn

Move

\frac{π}{6}

to the right side

θ - \frac{π}{6} = \frac{11 π}{6} + 2 πn

Add

\frac{π}{6}

to both sides

θ - \frac{π}{6} + \frac{π}{6} = \frac{11 π}{6} + 2 πn + \frac{π}{6}

Simplify

θ - \frac{π}{6} + \frac{π}{6} = \frac{11 π}{6} + 2 πn + \frac{π}{6}

Simplify

θ - \frac{π}{6} + \frac{π}{6} : θ

θ - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} = 0

= θ

Simplify

\frac{11 π}{6} + 2 πn + \frac{π}{6} : 2 πn + 2 π

\frac{11 π}{6} + 2 πn + \frac{π}{6}

Group like terms

= 2 πn + \frac{π}{6} + \frac{11 π}{6}

Combine the fractions

\frac{π}{6} + \frac{11 π}{6} : 2 π

Apply rule

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

= \frac{π + 11 π}{6}

Add similar elements:

π + 11 π = 12 π

= \frac{12 π}{6}

Divide the numbers:

\frac{12}{6} = 2

= 2 π

= 2 πn + 2 π

θ = 2 πn + 2 π

θ = 2 πn + 2 π

θ = 2 πn + 2 π

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

Popular Examples

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csc (θ) = - \frac{5}{3}

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2 sin^{2} (t) cos (t) = 0

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2 cos (θ) + sin (θ) = 0

2sin^2(x)+sin(x)-1=0,0<x<2pi

2 sin^{2} (x) + sin (x) - 1 = 0, 0 < x < 2 π

5sin(x)-12cos(x)=13

5 sin (x) - 12 cos (x) = 13

Frequently Asked Questions (FAQ)

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