What is the general solution for 2cos(pi/5 θ)=sqrt(3) ?

The general solution for 2cos(pi/5 θ)=sqrt(3) is θ= 5/6+10n,θ= 55/6+10n

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2cos(pi/5 θ)=sqrt(3)

Solution

2 cos (\frac{π}{5} θ) = 3

Solution

θ = \frac{5}{6} + 10 n, θ = \frac{55}{6} + 10 n

Degrees

θ = 47.74648 \dots^{\circ} + 572.95779 \dots^{\circ} n, θ = 525.21131 \dots^{\circ} + 572.95779 \dots^{\circ} n

Solution steps

2 cos (\frac{π}{5} θ) = 3

Divide both sides by

2

2 cos (\frac{π}{5} θ) = 3

Divide both sides by

2

\frac{2 cos ( \frac{π}{5} θ )}{2} = \frac{3}{2}

Simplify

cos (\frac{π}{5} θ) = \frac{3}{2}

cos (\frac{π}{5} θ) = \frac{3}{2}

General solutions for

cos (\frac{π}{5} 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}

\frac{π}{5} θ = \frac{π}{6} + 2 πn, \frac{π}{5} θ = \frac{11 π}{6} + 2 πn

\frac{π}{5} θ = \frac{π}{6} + 2 πn, \frac{π}{5} θ = \frac{11 π}{6} + 2 πn

Solve

\frac{π}{5} θ = \frac{π}{6} + 2 πn : θ = \frac{5}{6} + 10 n

\frac{π}{5} θ = \frac{π}{6} + 2 πn

Multiply both sides by

5

\frac{π}{5} θ = \frac{π}{6} + 2 πn

Multiply both sides by

5

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

Simplify

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

Simplify

5 \cdot \frac{π}{5} θ : π θ

5 \cdot \frac{π}{5} θ

Multiply fractions:

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

= \frac{5 π}{5} θ

Cancel the common factor:

5

= θ π

Simplify

5 \cdot \frac{π}{6} + 5 \cdot 2 πn : \frac{5 π}{6} + 10 πn

5 \cdot \frac{π}{6} + 5 \cdot 2 πn

Multiply

5 \cdot \frac{π}{6} : \frac{5 π}{6}

5 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 5}{6}

= \frac{5 π}{6} + 5 \cdot 2 πn

Multiply the numbers:

5 \cdot 2 = 10

= \frac{5 π}{6} + 10 πn

π θ = \frac{5 π}{6} + 10 πn

π θ = \frac{5 π}{6} + 10 πn

π θ = \frac{5 π}{6} + 10 πn

Divide both sides by

π

π θ = \frac{5 π}{6} + 10 πn

Divide both sides by

π

\frac{π θ}{π} = \frac{\frac{5 π}{6}}{π} + \frac{10 πn}{π}

Simplify

\frac{π θ}{π} = \frac{\frac{5 π}{6}}{π} + \frac{10 πn}{π}

Simplify

\frac{π θ}{π} : θ

\frac{π θ}{π}

Cancel the common factor:

π

= θ

Simplify

\frac{\frac{5 π}{6}}{π} + \frac{10 πn}{π} : \frac{5}{6} + 10 n

\frac{\frac{5 π}{6}}{π} + \frac{10 πn}{π}

\frac{\frac{5 π}{6}}{π} = \frac{5}{6}

\frac{\frac{5 π}{6}}{π}

Apply the fraction rule:

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

= \frac{5 π}{6 π}

Cancel the common factor:

π

= \frac{5}{6}

\frac{10 πn}{π} = 10 n

\frac{10 πn}{π}

Cancel the common factor:

π

= 10 n

= \frac{5}{6} + 10 n

θ = \frac{5}{6} + 10 n

θ = \frac{5}{6} + 10 n

θ = \frac{5}{6} + 10 n

Solve

\frac{π}{5} θ = \frac{11 π}{6} + 2 πn : θ = \frac{55}{6} + 10 n

\frac{π}{5} θ = \frac{11 π}{6} + 2 πn

Multiply both sides by

5

\frac{π}{5} θ = \frac{11 π}{6} + 2 πn

Multiply both sides by

5

5 \cdot \frac{π}{5} θ = 5 \cdot \frac{11 π}{6} + 5 \cdot 2 πn

Simplify

5 \cdot \frac{π}{5} θ = 5 \cdot \frac{11 π}{6} + 5 \cdot 2 πn

Simplify

5 \cdot \frac{π}{5} θ : π θ

5 \cdot \frac{π}{5} θ

Multiply fractions:

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

= \frac{5 π}{5} θ

Cancel the common factor:

5

= θ π

Simplify

5 \cdot \frac{11 π}{6} + 5 \cdot 2 πn : \frac{55 π}{6} + 10 πn

5 \cdot \frac{11 π}{6} + 5 \cdot 2 πn

5 \cdot \frac{11 π}{6} = \frac{55 π}{6}

5 \cdot \frac{11 π}{6}

Multiply fractions:

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

= \frac{11 π 5}{6}

Multiply the numbers:

11 \cdot 5 = 55

= \frac{55 π}{6}

5 \cdot 2 πn = 10 πn

5 \cdot 2 πn

Multiply the numbers:

5 \cdot 2 = 10

= 10 πn

= \frac{55 π}{6} + 10 πn

π θ = \frac{55 π}{6} + 10 πn

π θ = \frac{55 π}{6} + 10 πn

π θ = \frac{55 π}{6} + 10 πn

Divide both sides by

π

π θ = \frac{55 π}{6} + 10 πn

Divide both sides by

π

\frac{π θ}{π} = \frac{\frac{55 π}{6}}{π} + \frac{10 πn}{π}

Simplify

\frac{π θ}{π} = \frac{\frac{55 π}{6}}{π} + \frac{10 πn}{π}

Simplify

\frac{π θ}{π} : θ

\frac{π θ}{π}

Cancel the common factor:

π

= θ

Simplify

\frac{\frac{55 π}{6}}{π} + \frac{10 πn}{π} : \frac{55}{6} + 10 n

\frac{\frac{55 π}{6}}{π} + \frac{10 πn}{π}

\frac{\frac{55 π}{6}}{π} = \frac{55}{6}

\frac{\frac{55 π}{6}}{π}

Apply the fraction rule:

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

= \frac{55 π}{6 π}

Cancel the common factor:

π

= \frac{55}{6}

\frac{10 πn}{π} = 10 n

\frac{10 πn}{π}

Cancel the common factor:

π

= 10 n

= \frac{55}{6} + 10 n

θ = \frac{55}{6} + 10 n

θ = \frac{55}{6} + 10 n

θ = \frac{55}{6} + 10 n

θ = \frac{5}{6} + 10 n, θ = \frac{55}{6} + 10 n

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos(pi/5 θ)=sqrt(3) ?
The general solution for 2cos(pi/5 θ)=sqrt(3) is θ= 5/6+10n,θ= 55/6+10n