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

The general solution for tan(3x+(3pi)/4)=1 is x=(pin)/3-pi/6

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

Solution

tan (3 x + \frac{3 π}{4}) = 1

Solution

x = \frac{πn}{3} - \frac{π}{6}

Degrees

x = - 3 0^{\circ} + 6 0^{\circ} n

Solution steps

tan (3 x + \frac{3 π}{4}) = 1

General solutions for

tan (3 x + \frac{3 π}{4}) = 1

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}

3 x + \frac{3 π}{4} = \frac{π}{4} + πn

3 x + \frac{3 π}{4} = \frac{π}{4} + πn

Solve

3 x + \frac{3 π}{4} = \frac{π}{4} + πn : x = \frac{πn}{3} - \frac{π}{6}

3 x + \frac{3 π}{4} = \frac{π}{4} + πn

Move

\frac{3 π}{4}

to the right side

3 x + \frac{3 π}{4} = \frac{π}{4} + πn

Subtract

\frac{3 π}{4}

from both sides

3 x + \frac{3 π}{4} - \frac{3 π}{4} = \frac{π}{4} + πn - \frac{3 π}{4}

Simplify

3 x + \frac{3 π}{4} - \frac{3 π}{4} = \frac{π}{4} + πn - \frac{3 π}{4}

Simplify

3 x + \frac{3 π}{4} - \frac{3 π}{4} : 3 x

3 x + \frac{3 π}{4} - \frac{3 π}{4}

Add similar elements:

\frac{3 π}{4} - \frac{3 π}{4} = 0

= 3 x

Simplify

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

\frac{π}{4} + πn - \frac{3 π}{4}

Group like terms

= πn + \frac{π}{4} - \frac{3 π}{4}

Combine the fractions

\frac{π}{4} - \frac{3 π}{4} : - \frac{π}{2}

Apply rule

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

= \frac{π - 3 π}{4}

Add similar elements:

π - 3 π = - 2 π

= \frac{- 2 π}{4}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 π}{4}

Cancel the common factor:

2

= - \frac{π}{2}

= πn - \frac{π}{2}

3 x = πn - \frac{π}{2}

3 x = πn - \frac{π}{2}

3 x = πn - \frac{π}{2}

Divide both sides by

3

3 x = πn - \frac{π}{2}

Divide both sides by

3

\frac{3 x}{3} = \frac{πn}{3} - \frac{\frac{π}{2}}{3}

Simplify

\frac{3 x}{3} = \frac{πn}{3} - \frac{\frac{π}{2}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

\frac{πn}{3} - \frac{\frac{π}{2}}{3}

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{πn}{3} - \frac{π}{6}

x = \frac{πn}{3} - \frac{π}{6}

x = \frac{πn}{3} - \frac{π}{6}

x = \frac{πn}{3} - \frac{π}{6}

x = \frac{πn}{3} - \frac{π}{6}

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Frequently Asked Questions (FAQ)

What is the general solution for tan(3x+(3pi)/4)=1 ?
The general solution for tan(3x+(3pi)/4)=1 is x=(pin)/3-pi/6