What is the general solution for tan(3x)*tan(30)=1 ?

The general solution for tan(3x)*tan(30)=1 is x=20+(180n)/3

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tan(3x)*tan(30)=1

Solution

tan (3 x) \cdot tan (3 0^{\circ}) = 1

Solution

x = 2 0^{\circ} + \frac{18 0 ^{\circ} n}{3}

Radians

x = \frac{π}{9} + \frac{π}{3} n

Solution steps

tan (3 x) tan (3 0^{\circ}) = 1

tan (3 0^{\circ}) = \frac{3}{3}

tan (3 0^{\circ})

Use the following trivial identity:

tan (3 0^{\circ}) = \frac{3}{3}

tan (3 0^{\circ})

tan (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= \frac{3}{3}

= \frac{3}{3}

tan (3 x) \frac{3}{3} = 1

Multiply both sides by

3

tan (3 x) \frac{3}{3} = 1

Multiply both sides by

3

3 tan (3 x) \frac{3}{3} = 1 \cdot 3

Simplify

3 tan (3 x) = 3

3 tan (3 x) = 3

Divide both sides by

3

3 tan (3 x) = 3

Divide both sides by

3

\frac{3 tan ( 3 x )}{3} = \frac{3}{3}

Simplify

\frac{3 tan ( 3 x )}{3} = \frac{3}{3}

Simplify

\frac{3 tan ( 3 x )}{3} : tan (3 x)

\frac{3 tan ( 3 x )}{3}

Cancel the common factor:

3

= tan (3 x)

Simplify

\frac{3}{3} : 3

\frac{3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3}{3 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= 3^{1 - \frac{1}{2}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 3^{\frac{1}{2}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= 3

tan (3 x) = 3

tan (3 x) = 3

tan (3 x) = 3

General solutions for

tan (3 x) = 3

tan (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

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

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

Solve

3 x = 6 0^{\circ} + 18 0^{\circ} n : x = 2 0^{\circ} + \frac{18 0 ^{\circ} n}{3}

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

Divide both sides by

3

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

Divide both sides by

3

\frac{3 x}{3} = \frac{6 0 ^{\circ}}{3} + \frac{18 0 ^{\circ} n}{3}

Simplify

\frac{3 x}{3} = \frac{6 0 ^{\circ}}{3} + \frac{18 0 ^{\circ} n}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{6 0 ^{\circ}}{3} + \frac{18 0 ^{\circ} n}{3} : 2 0^{\circ} + \frac{18 0 ^{\circ} n}{3}

\frac{6 0 ^{\circ}}{3} + \frac{18 0 ^{\circ} n}{3}

\frac{6 0 ^{\circ}}{3} = 2 0^{\circ}

\frac{6 0 ^{\circ}}{3}

Apply the fraction rule:

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

= \frac{18 0 ^{\circ}}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= 2 0^{\circ}

= 2 0^{\circ} + \frac{18 0 ^{\circ} n}{3}

x = 2 0^{\circ} + \frac{18 0 ^{\circ} n}{3}

x = 2 0^{\circ} + \frac{18 0 ^{\circ} n}{3}

x = 2 0^{\circ} + \frac{18 0 ^{\circ} n}{3}

x = 2 0^{\circ} + \frac{18 0 ^{\circ} n}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for tan(3x)*tan(30)=1 ?
The general solution for tan(3x)*tan(30)=1 is x=20+(180n)/3