What is the general solution for sqrt(3)tan(x-pi/5)-1=0 ?

The general solution for sqrt(3)tan(x-pi/5)-1=0 is x=pin+(11pi)/(30)

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Popular Trigonometry >

sqrt(3)tan(x-pi/5)-1=0

Solution

3 tan (x - \frac{π}{5}) - 1 = 0

Solution

x = πn + \frac{11 π}{30}

Degrees

x = 6 6^{\circ} + 18 0^{\circ} n

Solution steps

3 tan (x - \frac{π}{5}) - 1 = 0

Move

1

to the right side

3 tan (x - \frac{π}{5}) - 1 = 0

Add

1

to both sides

3 tan (x - \frac{π}{5}) - 1 + 1 = 0 + 1

Simplify

3 tan (x - \frac{π}{5}) = 1

3 tan (x - \frac{π}{5}) = 1

Divide both sides by

3

3 tan (x - \frac{π}{5}) = 1

Divide both sides by

3

\frac{3 tan ( x - \frac{π}{5} )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( x - \frac{π}{5} )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( x - \frac{π}{5} )}{3} : tan (x - \frac{π}{5})

\frac{3 tan ( x - \frac{π}{5} )}{3}

Cancel the common factor:

3

= tan (x - \frac{π}{5})

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 (x - \frac{π}{5}) = \frac{3}{3}

tan (x - \frac{π}{5}) = \frac{3}{3}

tan (x - \frac{π}{5}) = \frac{3}{3}

General solutions for

tan (x - \frac{π}{5}) = \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}

x - \frac{π}{5} = \frac{π}{6} + πn

x - \frac{π}{5} = \frac{π}{6} + πn

Solve

x - \frac{π}{5} = \frac{π}{6} + πn : x = πn + \frac{11 π}{30}

x - \frac{π}{5} = \frac{π}{6} + πn

Move

\frac{π}{5}

to the right side

x - \frac{π}{5} = \frac{π}{6} + πn

Add

\frac{π}{5}

to both sides

x - \frac{π}{5} + \frac{π}{5} = \frac{π}{6} + πn + \frac{π}{5}

Simplify

x - \frac{π}{5} + \frac{π}{5} = \frac{π}{6} + πn + \frac{π}{5}

Simplify

x - \frac{π}{5} + \frac{π}{5} : x

x - \frac{π}{5} + \frac{π}{5}

Add similar elements:

- \frac{π}{5} + \frac{π}{5} = 0

= x

Simplify

\frac{π}{6} + πn + \frac{π}{5} : πn + \frac{11 π}{30}

\frac{π}{6} + πn + \frac{π}{5}

Group like terms

= πn + \frac{π}{6} + \frac{π}{5}

Least Common Multiplier of

6, 5 : 30

6, 5

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

5 : 5

5

5

is a prime number, therefore no factorization is possible

= 5

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

6

5

= 2 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 3 \cdot 5 = 30

= 30

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

30

For

\frac{π}{6} :

multiply the denominator and numerator by

5

\frac{π}{6} = \frac{π 5}{6 \cdot 5} = \frac{π 5}{30}

For

\frac{π}{5} :

multiply the denominator and numerator by

6

\frac{π}{5} = \frac{π 6}{5 \cdot 6} = \frac{π 6}{30}

= \frac{π 5}{30} + \frac{π 6}{30}

Since the denominators are equal, combine the fractions:

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

= \frac{π 5 + π 6}{30}

Add similar elements:

5 π + 6 π = 11 π

= πn + \frac{11 π}{30}

x = πn + \frac{11 π}{30}

x = πn + \frac{11 π}{30}

x = πn + \frac{11 π}{30}

x = πn + \frac{11 π}{30}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)tan(x-pi/5)-1=0 ?
The general solution for sqrt(3)tan(x-pi/5)-1=0 is x=pin+(11pi)/(30)