What is the general solution for tan(5x)tan(x)=tan(5x) ?

The general solution for tan(5x)tan(x)=tan(5x) is x=(pin)/5 ,x= pi/4+pin

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tan(5x)tan(x)=tan(5x)

Solution

tan (5 x) tan (x) = tan (5 x)

Solution

x = \frac{πn}{5}, x = \frac{π}{4} + πn

Degrees

x = 0^{\circ} + 3 6^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

tan (5 x) tan (x) = tan (5 x)

Subtract

tan (5 x)

from both sides

tan (5 x) tan (x) - tan (5 x) = 0

Factor

tan (5 x) tan (x) - tan (5 x) : tan (5 x) (tan (x) - 1)

tan (5 x) tan (x) - tan (5 x)

Factor out common term

tan (5 x)

= tan (5 x) (tan (x) - 1)

tan (5 x) (tan (x) - 1) = 0

Solving each part separately

tan (5 x) = 0 or tan (x) - 1 = 0

tan (5 x) = 0 : x = \frac{πn}{5}

tan (5 x) = 0

General solutions for

tan (5 x) = 0

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}

5 x = 0 + πn

5 x = 0 + πn

Solve

5 x = 0 + πn : x = \frac{πn}{5}

5 x = 0 + πn

0 + πn = πn

5 x = πn

Divide both sides by

5

5 x = πn

Divide both sides by

5

\frac{5 x}{5} = \frac{πn}{5}

Simplify

x = \frac{πn}{5}

x = \frac{πn}{5}

x = \frac{πn}{5}

tan (x) - 1 = 0 : x = \frac{π}{4} + πn

tan (x) - 1 = 0

Move

1

to the right side

tan (x) - 1 = 0

Add

1

to both sides

tan (x) - 1 + 1 = 0 + 1

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

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}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = \frac{πn}{5}, x = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan(5x)tan(x)=tan(5x) ?
The general solution for tan(5x)tan(x)=tan(5x) is x=(pin)/5 ,x= pi/4+pin