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

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

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tan(x-pi/2)= 1/(sqrt(3))

Solution

tan (x - \frac{π}{2}) = \frac{1}{3}

Solution

x = πn + \frac{2 π}{3}

Radians

x = \frac{2 π}{3} + πn

Solution steps

tan (x - \frac{π}{2}) = \frac{1}{3}

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}{2}) = \frac{3}{3}

General solutions for

tan (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}

x - \frac{π}{2} = \frac{π}{6} + πn

x - \frac{π}{2} = \frac{π}{6} + πn

Solve

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

x - \frac{π}{2} = \frac{π}{6} + πn

Move

\frac{π}{2}

to the right side

x - \frac{π}{2} = \frac{π}{6} + πn

Add

\frac{π}{2}

to both sides

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

Simplify

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

Simplify

x - \frac{π}{2} + \frac{π}{2} : x

x - \frac{π}{2} + \frac{π}{2}

Add similar elements:

- \frac{π}{2} + \frac{π}{2} = 0

= x

Simplify

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

\frac{π}{6} + πn + \frac{π}{2}

Group like terms

= πn + \frac{π}{6} + \frac{π}{2}

Least Common Multiplier of

6, 2 : 6

6, 2

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

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

6

2

= 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}{6} + \frac{π 3}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π + π 3}{6}

Add similar elements:

π + 3 π = 4 π

= \frac{4 π}{6}

Cancel the common factor:

2

= πn + \frac{2 π}{3}

x = πn + \frac{2 π}{3}

x = πn + \frac{2 π}{3}

x = πn + \frac{2 π}{3}

x = πn + \frac{2 π}{3}

Popular Examples

2cos(x)=tan(x)

2 cos (x) = tan (x)

sin(x)+sin^2(x)=1

sin (x) + sin^{2} (x) = 1

sin^3(a)=3sin(a)-4sin^3(a)

sin^{3} (a) = 3 sin (a) - 4 sin^{3} (a)

cos(3x)= 3/2

cos (3 x) = \frac{3}{2}

sin(θ)+2sin(θ)cos(θ)=0

sin (θ) + 2 sin (θ) cos (θ) = 0

Frequently Asked Questions (FAQ)

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