What is the general solution for cos((5x}{10})=\frac{sqrt(2))/2 ?

The general solution for cos((5x}{10})=\frac{sqrt(2))/2 is x= pi/2+4pin,x=(7pi)/2+4pin

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cos((5x}{10})=\frac{sqrt(2))/2

Solution

cos (\frac{5 x}{10}) = \frac{2}{2}

Solution

x = \frac{π}{2} + 4 πn, x = \frac{7 π}{2} + 4 πn

Degrees

x = 9 0^{\circ} + 72 0^{\circ} n, x = 63 0^{\circ} + 72 0^{\circ} n

Solution steps

cos (\frac{5 x}{10}) = \frac{2}{2}

General solutions for

cos (\frac{5 x}{10}) = \frac{2}{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{5 x}{10} = \frac{π}{4} + 2 πn, \frac{5 x}{10} = \frac{7 π}{4} + 2 πn

\frac{5 x}{10} = \frac{π}{4} + 2 πn, \frac{5 x}{10} = \frac{7 π}{4} + 2 πn

Solve

\frac{5 x}{10} = \frac{π}{4} + 2 πn : x = \frac{π}{2} + 4 πn

\frac{5 x}{10} = \frac{π}{4} + 2 πn

Multiply both sides by

10

\frac{5 x}{10} = \frac{π}{4} + 2 πn

Multiply both sides by

10

\frac{10 \cdot 5 x}{10} = 10 \cdot \frac{π}{4} + 10 \cdot 2 πn

Simplify

\frac{10 \cdot 5 x}{10} = 10 \cdot \frac{π}{4} + 10 \cdot 2 πn

Simplify

\frac{10 \cdot 5 x}{10} : 5 x

\frac{10 \cdot 5 x}{10}

Multiply the numbers:

10 \cdot 5 = 50

= \frac{50 x}{10}

Divide the numbers:

\frac{50}{10} = 5

= 5 x

Simplify

10 \cdot \frac{π}{4} + 10 \cdot 2 πn : \frac{5 π}{2} + 20 πn

10 \cdot \frac{π}{4} + 10 \cdot 2 πn

10 \cdot \frac{π}{4} = \frac{5 π}{2}

10 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 10}{4}

Cancel the common factor:

2

= \frac{5 π}{2}

10 \cdot 2 πn = 20 πn

10 \cdot 2 πn

Multiply the numbers:

10 \cdot 2 = 20

= 20 πn

= \frac{5 π}{2} + 20 πn

5 x = \frac{5 π}{2} + 20 πn

5 x = \frac{5 π}{2} + 20 πn

5 x = \frac{5 π}{2} + 20 πn

Divide both sides by

5

5 x = \frac{5 π}{2} + 20 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{\frac{5 π}{2}}{5} + \frac{20 πn}{5}

Simplify

\frac{5 x}{5} = \frac{\frac{5 π}{2}}{5} + \frac{20 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{5 π}{2}}{5} + \frac{20 πn}{5} : \frac{π}{2} + 4 πn

\frac{\frac{5 π}{2}}{5} + \frac{20 πn}{5}

\frac{\frac{5 π}{2}}{5} = \frac{π}{2}

\frac{\frac{5 π}{2}}{5}

Apply the fraction rule:

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

= \frac{5 π}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{5 π}{10}

Cancel the common factor:

5

= \frac{π}{2}

\frac{20 πn}{5} = 4 πn

\frac{20 πn}{5}

Divide the numbers:

\frac{20}{5} = 4

= 4 πn

= \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

Solve

\frac{5 x}{10} = \frac{7 π}{4} + 2 πn : x = \frac{7 π}{2} + 4 πn

\frac{5 x}{10} = \frac{7 π}{4} + 2 πn

Multiply both sides by

10

\frac{5 x}{10} = \frac{7 π}{4} + 2 πn

Multiply both sides by

10

\frac{10 \cdot 5 x}{10} = 10 \cdot \frac{7 π}{4} + 10 \cdot 2 πn

Simplify

\frac{10 \cdot 5 x}{10} = 10 \cdot \frac{7 π}{4} + 10 \cdot 2 πn

Simplify

\frac{10 \cdot 5 x}{10} : 5 x

\frac{10 \cdot 5 x}{10}

Multiply the numbers:

10 \cdot 5 = 50

= \frac{50 x}{10}

Divide the numbers:

\frac{50}{10} = 5

= 5 x

Simplify

10 \cdot \frac{7 π}{4} + 10 \cdot 2 πn : \frac{35 π}{2} + 20 πn

10 \cdot \frac{7 π}{4} + 10 \cdot 2 πn

10 \cdot \frac{7 π}{4} = \frac{35 π}{2}

10 \cdot \frac{7 π}{4}

Multiply fractions:

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

= \frac{7 π 10}{4}

Multiply the numbers:

7 \cdot 10 = 70

= \frac{70 π}{4}

Cancel the common factor:

2

= \frac{35 π}{2}

10 \cdot 2 πn = 20 πn

10 \cdot 2 πn

Multiply the numbers:

10 \cdot 2 = 20

= 20 πn

= \frac{35 π}{2} + 20 πn

5 x = \frac{35 π}{2} + 20 πn

5 x = \frac{35 π}{2} + 20 πn

5 x = \frac{35 π}{2} + 20 πn

Divide both sides by

5

5 x = \frac{35 π}{2} + 20 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{\frac{35 π}{2}}{5} + \frac{20 πn}{5}

Simplify

\frac{5 x}{5} = \frac{\frac{35 π}{2}}{5} + \frac{20 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{35 π}{2}}{5} + \frac{20 πn}{5} : \frac{7 π}{2} + 4 πn

\frac{\frac{35 π}{2}}{5} + \frac{20 πn}{5}

\frac{\frac{35 π}{2}}{5} = \frac{7 π}{2}

\frac{\frac{35 π}{2}}{5}

Apply the fraction rule:

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

= \frac{35 π}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{35 π}{10}

Cancel the common factor:

5

= \frac{7 π}{2}

\frac{20 πn}{5} = 4 πn

\frac{20 πn}{5}

Divide the numbers:

\frac{20}{5} = 4

= 4 πn

= \frac{7 π}{2} + 4 πn

x = \frac{7 π}{2} + 4 πn

x = \frac{7 π}{2} + 4 πn

x = \frac{7 π}{2} + 4 πn

x = \frac{π}{2} + 4 πn, x = \frac{7 π}{2} + 4 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos((5x}{10})=\frac{sqrt(2))/2 ?
The general solution for cos((5x}{10})=\frac{sqrt(2))/2 is x= pi/2+4pin,x=(7pi)/2+4pin