What is the general solution for -1=tan(x+pi/(18)) ?

The general solution for -1=tan(x+pi/(18)) is x=pin+(25pi)/(36)

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Popular Trigonometry >

-1=tan(x+pi/(18))

Solution

- 1 = tan (x + \frac{π}{18})

Solution

x = πn + \frac{25 π}{36}

Degrees

x = 12 5^{\circ} + 18 0^{\circ} n

Solution steps

- 1 = tan (x + \frac{π}{18})

Switch sides

tan (x + \frac{π}{18}) = - 1

General solutions for

tan (x + \frac{π}{18}) = - 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}{18} = \frac{3 π}{4} + πn

x + \frac{π}{18} = \frac{3 π}{4} + πn

Solve

x + \frac{π}{18} = \frac{3 π}{4} + πn : x = πn + \frac{25 π}{36}

x + \frac{π}{18} = \frac{3 π}{4} + πn

Move

\frac{π}{18}

to the right side

x + \frac{π}{18} = \frac{3 π}{4} + πn

Subtract

\frac{π}{18}

from both sides

x + \frac{π}{18} - \frac{π}{18} = \frac{3 π}{4} + πn - \frac{π}{18}

Simplify

x + \frac{π}{18} - \frac{π}{18} = \frac{3 π}{4} + πn - \frac{π}{18}

Simplify

x + \frac{π}{18} - \frac{π}{18} : x

x + \frac{π}{18} - \frac{π}{18}

Add similar elements:

\frac{π}{18} - \frac{π}{18} = 0

= x

Simplify

\frac{3 π}{4} + πn - \frac{π}{18} : πn + \frac{25 π}{36}

\frac{3 π}{4} + πn - \frac{π}{18}

Group like terms

= πn + \frac{3 π}{4} - \frac{π}{18}

Least Common Multiplier of

4, 18 : 36

4, 18

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

18 : 2 \cdot 3 \cdot 3

18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3

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

4

18

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

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

36

For

\frac{3 π}{4} :

multiply the denominator and numerator by

9

\frac{3 π}{4} = \frac{3 π 9}{4 \cdot 9} = \frac{27 π}{36}

For

\frac{π}{18} :

multiply the denominator and numerator by

2

\frac{π}{18} = \frac{π 2}{18 \cdot 2} = \frac{π 2}{36}

= \frac{27 π}{36} - \frac{π 2}{36}

Since the denominators are equal, combine the fractions:

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

= \frac{27 π - π 2}{36}

Add similar elements:

27 π - 2 π = 25 π

= πn + \frac{25 π}{36}

x = πn + \frac{25 π}{36}

x = πn + \frac{25 π}{36}

x = πn + \frac{25 π}{36}

x = πn + \frac{25 π}{36}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for -1=tan(x+pi/(18)) ?
The general solution for -1=tan(x+pi/(18)) is x=pin+(25pi)/(36)