What is the general solution for 8tan(-x+pi/8)=8 ?

The general solution for 8tan(-x+pi/8)=8 is x=-pin-pi/8

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8tan(-x+pi/8)=8

Solution

8 tan (- x + \frac{π}{8}) = 8

Solution

x = - πn - \frac{π}{8}

Degrees

x = - 22. 5^{\circ} - 18 0^{\circ} n

Solution steps

8 tan (- x + \frac{π}{8}) = 8

Divide both sides by

8

8 tan (- x + \frac{π}{8}) = 8

Divide both sides by

8

\frac{8 tan ( - x + \frac{π}{8} )}{8} = \frac{8}{8}

Simplify

tan (- x + \frac{π}{8}) = 1

tan (- x + \frac{π}{8}) = 1

General solutions for

tan (- x + \frac{π}{8}) = 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}

- x + \frac{π}{8} = \frac{π}{4} + πn

- x + \frac{π}{8} = \frac{π}{4} + πn

Solve

- x + \frac{π}{8} = \frac{π}{4} + πn : x = - πn - \frac{π}{8}

- x + \frac{π}{8} = \frac{π}{4} + πn

Move

\frac{π}{8}

to the right side

- x + \frac{π}{8} = \frac{π}{4} + πn

Subtract

\frac{π}{8}

from both sides

- x + \frac{π}{8} - \frac{π}{8} = \frac{π}{4} + πn - \frac{π}{8}

Simplify

- x + \frac{π}{8} - \frac{π}{8} = \frac{π}{4} + πn - \frac{π}{8}

Simplify

- x + \frac{π}{8} - \frac{π}{8} : - x

- x + \frac{π}{8} - \frac{π}{8}

Add similar elements:

\frac{π}{8} - \frac{π}{8} = 0

= - x

Simplify

\frac{π}{4} + πn - \frac{π}{8} : πn + \frac{π}{8}

\frac{π}{4} + πn - \frac{π}{8}

Group like terms

= πn + \frac{π}{4} - \frac{π}{8}

Least Common Multiplier of

4, 8 : 8

4, 8

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

4

8

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

8

For

\frac{π}{4} :

multiply the denominator and numerator by

2

\frac{π}{4} = \frac{π 2}{4 \cdot 2} = \frac{π 2}{8}

= \frac{π 2}{8} - \frac{π}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - π}{8}

Add similar elements:

2 π - π = π

= πn + \frac{π}{8}

- x = πn + \frac{π}{8}

- x = πn + \frac{π}{8}

- x = πn + \frac{π}{8}

Divide both sides by

- 1

- x = πn + \frac{π}{8}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{πn}{- 1} + \frac{\frac{π}{8}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{πn}{- 1} + \frac{\frac{π}{8}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{πn}{- 1} + \frac{\frac{π}{8}}{- 1} : - πn - \frac{π}{8}

\frac{πn}{- 1} + \frac{\frac{π}{8}}{- 1}

\frac{πn}{- 1} = - πn

\frac{πn}{- 1}

Apply the fraction rule:

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

= - \frac{πn}{1}

Apply rule

\frac{a}{1} = a

= - πn

= - πn + \frac{\frac{π}{8}}{- 1}

\frac{\frac{π}{8}}{- 1} = - \frac{π}{8}

\frac{\frac{π}{8}}{- 1}

Apply the fraction rule:

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

= - \frac{\frac{π}{8}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{\frac{π}{8}}{1} = \frac{π}{8}

= - \frac{π}{8}

= - πn - \frac{π}{8}

x = - πn - \frac{π}{8}

x = - πn - \frac{π}{8}

x = - πn - \frac{π}{8}

x = - πn - \frac{π}{8}

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Frequently Asked Questions (FAQ)

What is the general solution for 8tan(-x+pi/8)=8 ?
The general solution for 8tan(-x+pi/8)=8 is x=-pin-pi/8