What is the general solution for sqrt(3)tan(3x+pi/2)+1=0 ?

The general solution for sqrt(3)tan(3x+pi/2)+1=0 is x=(pin)/3+pi/9

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Popular Trigonometry >

sqrt(3)tan(3x+pi/2)+1=0

Solution

3 tan (3 x + \frac{π}{2}) + 1 = 0

Solution

x = \frac{πn}{3} + \frac{π}{9}

Degrees

x = 2 0^{\circ} + 6 0^{\circ} n

Solution steps

3 tan (3 x + \frac{π}{2}) + 1 = 0

Move

1

to the right side

3 tan (3 x + \frac{π}{2}) + 1 = 0

Subtract

1

from both sides

3 tan (3 x + \frac{π}{2}) + 1 - 1 = 0 - 1

Simplify

3 tan (3 x + \frac{π}{2}) = - 1

3 tan (3 x + \frac{π}{2}) = - 1

Divide both sides by

3

3 tan (3 x + \frac{π}{2}) = - 1

Divide both sides by

3

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

Simplify

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

Simplify

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

\frac{3 tan ( 3 x + \frac{π}{2} )}{3}

Cancel the common factor:

3

= tan (3 x + \frac{π}{2})

Simplify

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

\frac{- 1}{3}

Apply the fraction rule:

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

= - \frac{1}{3}

Rationalize

- \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}

= - \frac{3}{3}

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

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

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

General solutions for

tan (3 x + \frac{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}

3 x + \frac{π}{2} = \frac{5 π}{6} + πn

3 x + \frac{π}{2} = \frac{5 π}{6} + πn

Solve

3 x + \frac{π}{2} = \frac{5 π}{6} + πn : x = \frac{πn}{3} + \frac{π}{9}

3 x + \frac{π}{2} = \frac{5 π}{6} + πn

Move

\frac{π}{2}

to the right side

3 x + \frac{π}{2} = \frac{5 π}{6} + πn

Subtract

\frac{π}{2}

from both sides

3 x + \frac{π}{2} - \frac{π}{2} = \frac{5 π}{6} + πn - \frac{π}{2}

Simplify

3 x + \frac{π}{2} - \frac{π}{2} = \frac{5 π}{6} + πn - \frac{π}{2}

Simplify

3 x + \frac{π}{2} - \frac{π}{2} : 3 x

3 x + \frac{π}{2} - \frac{π}{2}

Add similar elements:

\frac{π}{2} - \frac{π}{2} = 0

= 3 x

Simplify

\frac{5 π}{6} + πn - \frac{π}{2} : πn + \frac{π}{3}

\frac{5 π}{6} + πn - \frac{π}{2}

Group like terms

= πn - \frac{π}{2} + \frac{5 π}{6}

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

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

2

6

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

= - \frac{π 3}{6} + \frac{5 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + 5 π}{6}

Add similar elements:

- 3 π + 5 π = 2 π

= \frac{2 π}{6}

Cancel the common factor:

2

= πn + \frac{π}{3}

3 x = πn + \frac{π}{3}

3 x = πn + \frac{π}{3}

3 x = πn + \frac{π}{3}

Divide both sides by

3

3 x = πn + \frac{π}{3}

Divide both sides by

3

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

Simplify

\frac{3 x}{3} = \frac{πn}{3} + \frac{\frac{π}{3}}{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}{3}}{3} : \frac{πn}{3} + \frac{π}{9}

\frac{πn}{3} + \frac{\frac{π}{3}}{3}

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= \frac{πn}{3} + \frac{π}{9}

x = \frac{πn}{3} + \frac{π}{9}

x = \frac{πn}{3} + \frac{π}{9}

x = \frac{πn}{3} + \frac{π}{9}

x = \frac{πn}{3} + \frac{π}{9}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)tan(3x+pi/2)+1=0 ?
The general solution for sqrt(3)tan(3x+pi/2)+1=0 is x=(pin)/3+pi/9