What is the general solution for sqrt(3)tan((3pi)/4 (x+(3pi)/2))=1 ?

The general solution for sqrt(3)tan((3pi)/4 (x+(3pi)/2))=1 is x= 2/9-(3pi)/2+(4n)/3

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sqrt(3)tan((3pi)/4 (x+(3pi)/2))=1

Solution

3 tan (\frac{3 π}{4} (x + \frac{3 π}{2})) = 1

Solution

x = \frac{2}{9} - \frac{3 π}{2} + \frac{4 n}{3}

Degrees

x = - 257.26760 \dots^{\circ} + 76.39437 \dots^{\circ} n

Solution steps

3 tan (\frac{3 π}{4} (x + \frac{3 π}{2})) = 1

Divide both sides by

3

3 tan (\frac{3 π}{4} (x + \frac{3 π}{2})) = 1

Divide both sides by

3

\frac{3 tan ( \frac{3 π}{4} ( x + \frac{3 π}{2} ) )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( \frac{3 π}{4} ( x + \frac{3 π}{2} ) )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( \frac{3 π}{4} ( x + \frac{3 π}{2} ) )}{3} : tan (\frac{3 π}{4} (x + \frac{3 π}{2}))

\frac{3 tan ( \frac{3 π}{4} ( x + \frac{3 π}{2} ) )}{3}

Cancel the common factor:

3

= tan (\frac{3 π}{4} (x + \frac{3 π}{2}))

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

tan (\frac{3 π}{4} (x + \frac{3 π}{2})) = \frac{3}{3}

tan (\frac{3 π}{4} (x + \frac{3 π}{2})) = \frac{3}{3}

tan (\frac{3 π}{4} (x + \frac{3 π}{2})) = \frac{3}{3}

General solutions for

tan (\frac{3 π}{4} (x + \frac{3 π}{2})) = \frac{3}{3}

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

\frac{3 π}{4} (x + \frac{3 π}{2}) = \frac{π}{6} + πn

\frac{3 π}{4} (x + \frac{3 π}{2}) = \frac{π}{6} + πn

Solve

\frac{3 π}{4} (x + \frac{3 π}{2}) = \frac{π}{6} + πn : x = \frac{2}{9} - \frac{3 π}{2} + \frac{4 n}{3}

\frac{3 π}{4} (x + \frac{3 π}{2}) = \frac{π}{6} + πn

Multiply both sides by

4

\frac{3 π}{4} (x + \frac{3 π}{2}) = \frac{π}{6} + πn

Multiply both sides by

4

4 \cdot \frac{3 π}{4} (x + \frac{3 π}{2}) = 4 \cdot \frac{π}{6} + 4 πn

Simplify

4 \cdot \frac{3 π}{4} (x + \frac{3 π}{2}) = 4 \cdot \frac{π}{6} + 4 πn

Simplify

4 \cdot \frac{3 π}{4} (x + \frac{3 π}{2}) : 3 π (x + \frac{3 π}{2})

4 \cdot \frac{3 π}{4} (x + \frac{3 π}{2})

Multiply fractions:

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

= \frac{3 \cdot 4 π}{4} (x + \frac{3 π}{2})

Cancel the common factor:

4

= (x + \frac{3 π}{2}) \cdot 3 π

Simplify

4 \cdot \frac{π}{6} + 4 πn : \frac{2 π}{3} + 4 πn

4 \cdot \frac{π}{6} + 4 πn

4 \cdot \frac{π}{6} = \frac{2 π}{3}

4 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 4}{6}

Cancel the common factor:

2

= \frac{2 π}{3}

= \frac{2 π}{3} + 4 πn

3 π (x + \frac{3 π}{2}) = \frac{2 π}{3} + 4 πn

3 π (x + \frac{3 π}{2}) = \frac{2 π}{3} + 4 πn

3 π (x + \frac{3 π}{2}) = \frac{2 π}{3} + 4 πn

Divide both sides by

3 π

3 π (x + \frac{3 π}{2}) = \frac{2 π}{3} + 4 πn

Divide both sides by

3 π

\frac{3 π ( x + \frac{3 π}{2} )}{3 π} = \frac{\frac{2 π}{3}}{3 π} + \frac{4 πn}{3 π}

Simplify

\frac{3 π ( x + \frac{3 π}{2} )}{3 π} = \frac{\frac{2 π}{3}}{3 π} + \frac{4 πn}{3 π}

Simplify

\frac{3 π ( x + \frac{3 π}{2} )}{3 π} : x + \frac{3 π}{2}

\frac{3 π ( x + \frac{3 π}{2} )}{3 π}

Divide the numbers:

\frac{3}{3} = 1

= \frac{π ( x + \frac{3 π}{2} )}{π}

Cancel the common factor:

π

= x + \frac{3 π}{2}

Simplify

\frac{\frac{2 π}{3}}{3 π} + \frac{4 πn}{3 π} : \frac{2}{9} + \frac{4 n}{3}

\frac{\frac{2 π}{3}}{3 π} + \frac{4 πn}{3 π}

\frac{\frac{2 π}{3}}{3 π} = \frac{2}{9}

\frac{\frac{2 π}{3}}{3 π}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 3 π}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{2 π}{9 π}

Cancel the common factor:

π

= \frac{2}{9}

\frac{4 πn}{3 π} = \frac{4 n}{3}

\frac{4 πn}{3 π}

Cancel the common factor:

π

= \frac{4 n}{3}

= \frac{2}{9} + \frac{4 n}{3}

x + \frac{3 π}{2} = \frac{2}{9} + \frac{4 n}{3}

x + \frac{3 π}{2} = \frac{2}{9} + \frac{4 n}{3}

x + \frac{3 π}{2} = \frac{2}{9} + \frac{4 n}{3}

Move

\frac{3 π}{2}

to the right side

x + \frac{3 π}{2} = \frac{2}{9} + \frac{4 n}{3}

Subtract

\frac{3 π}{2}

from both sides

x + \frac{3 π}{2} - \frac{3 π}{2} = \frac{2}{9} + \frac{4 n}{3} - \frac{3 π}{2}

Simplify

x + \frac{3 π}{2} - \frac{3 π}{2} = \frac{2}{9} + \frac{4 n}{3} - \frac{3 π}{2}

Simplify

x + \frac{3 π}{2} - \frac{3 π}{2} : x

x + \frac{3 π}{2} - \frac{3 π}{2}

Add similar elements:

\frac{3 π}{2} - \frac{3 π}{2} = 0

= x

Simplify

\frac{2}{9} + \frac{4 n}{3} - \frac{3 π}{2} : \frac{2}{9} - \frac{3 π}{2} + \frac{4 n}{3}

\frac{2}{9} + \frac{4 n}{3} - \frac{3 π}{2}

Group like terms

= \frac{2}{9} - \frac{3 π}{2} + \frac{4 n}{3}

Could not simplify further

= \frac{2}{9} - \frac{3 π}{2} + \frac{4 n}{3}

x = \frac{2}{9} - \frac{3 π}{2} + \frac{4 n}{3}

x = \frac{2}{9} - \frac{3 π}{2} + \frac{4 n}{3}

x = \frac{2}{9} - \frac{3 π}{2} + \frac{4 n}{3}

x = \frac{2}{9} - \frac{3 π}{2} + \frac{4 n}{3}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)tan((3pi)/4 (x+(3pi)/2))=1 ?
The general solution for sqrt(3)tan((3pi)/4 (x+(3pi)/2))=1 is x= 2/9-(3pi)/2+(4n)/3