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

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

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3tan(2x-pi/2)=2sin(pi/3)

Solution

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

Solution

x = \frac{πn}{2} + \frac{π}{3}

Degrees

x = 6 0^{\circ} + 9 0^{\circ} n

Solution steps

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

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= \frac{3}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

3 tan (2 x - \frac{π}{2}) = 3

Divide both sides by

3

3 tan (2 x - \frac{π}{2}) = 3

Divide both sides by

3

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

Simplify

tan (2 x - \frac{π}{2}) = \frac{3}{3}

tan (2 x - \frac{π}{2}) = \frac{3}{3}

General solutions for

tan (2 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}

2 x - \frac{π}{2} = \frac{π}{6} + πn

2 x - \frac{π}{2} = \frac{π}{6} + πn

Solve

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

2 x - \frac{π}{2} = \frac{π}{6} + πn

Move

\frac{π}{2}

to the right side

2 x - \frac{π}{2} = \frac{π}{6} + πn

Add

\frac{π}{2}

to both sides

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

Simplify

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

Simplify

2 x - \frac{π}{2} + \frac{π}{2} : 2 x

2 x - \frac{π}{2} + \frac{π}{2}

Add similar elements:

- \frac{π}{2} + \frac{π}{2} = 0

= 2 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}

2 x = πn + \frac{2 π}{3}

2 x = πn + \frac{2 π}{3}

2 x = πn + \frac{2 π}{3}

Divide both sides by

2

2 x = πn + \frac{2 π}{3}

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

= \frac{πn}{2} + \frac{π}{3}

x = \frac{πn}{2} + \frac{π}{3}

x = \frac{πn}{2} + \frac{π}{3}

x = \frac{πn}{2} + \frac{π}{3}

x = \frac{πn}{2} + \frac{π}{3}

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Frequently Asked Questions (FAQ)

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