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

The general solution for (tan(3x))/(tan(2x))=1 is No Solution for x\in\mathbb{R}

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(tan(3x))/(tan(2x))=1

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

1

from both sides

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

Simplify

\frac{tan ( 3 x )}{tan ( 2 x )} - 1 : \frac{tan ( 3 x ) - tan ( 2 x )}{tan ( 2 x )}

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

Convert element to fraction:

1 = \frac{1 tan ( 2 x )}{tan ( 2 x )}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot tan (2 x) = tan (2 x)

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

tan (3 x) - tan (2 x) = 0

Express with sin, cos

- tan (2 x) + tan (3 x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

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

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - \frac{sin ( 2 x )}{cos ( 2 x )} + \frac{sin ( 3 x )}{cos ( 3 x )}

Simplify

- \frac{sin ( 2 x )}{cos ( 2 x )} + \frac{sin ( 3 x )}{cos ( 3 x )} : \frac{- sin ( 2 x ) cos ( 3 x ) + sin ( 3 x ) cos ( 2 x )}{cos ( 2 x ) cos ( 3 x )}

- \frac{sin ( 2 x )}{cos ( 2 x )} + \frac{sin ( 3 x )}{cos ( 3 x )}

Least Common Multiplier of

cos (2 x), cos (3 x) : cos (2 x) cos (3 x)

cos (2 x), cos (3 x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (2 x)

cos (3 x)

= cos (2 x) cos (3 x)

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

cos (2 x) cos (3 x)

For

\frac{sin ( 2 x )}{cos ( 2 x )} :

multiply the denominator and numerator by

cos (3 x)

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

For

\frac{sin ( 3 x )}{cos ( 3 x )} :

multiply the denominator and numerator by

cos (2 x)

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

= - \frac{sin ( 2 x ) cos ( 3 x )}{cos ( 2 x ) cos ( 3 x )} + \frac{sin ( 3 x ) cos ( 2 x )}{cos ( 3 x ) cos ( 2 x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- sin ( 2 x ) cos ( 3 x ) + sin ( 3 x ) cos ( 2 x )}{cos ( 2 x ) cos ( 3 x )}

= \frac{- sin ( 2 x ) cos ( 3 x ) + sin ( 3 x ) cos ( 2 x )}{cos ( 2 x ) cos ( 3 x )}

\frac{cos ( 2 x ) sin ( 3 x ) - cos ( 3 x ) sin ( 2 x )}{cos ( 2 x ) cos ( 3 x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos (2 x) sin (3 x) - cos (3 x) sin (2 x) = 0

Rewrite using trig identities

cos (2 x) sin (3 x) - cos (3 x) sin (2 x)

Use the Angle Difference identity:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= sin (3 x - 2 x)

sin (3 x - 2 x) = 0

General solutions for

sin (3 x - 2 x) = 0

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}

3 x - 2 x = 0 + 2 πn, 3 x - 2 x = π + 2 πn

3 x - 2 x = 0 + 2 πn, 3 x - 2 x = π + 2 πn

Solve

3 x - 2 x = 0 + 2 πn : x = 2 πn

3 x - 2 x = 0 + 2 πn

Add similar elements:

3 x - 2 x = x

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

Solve

3 x - 2 x = π + 2 πn : x = π + 2 πn

3 x - 2 x = π + 2 πn

Add similar elements:

3 x - 2 x = x

x = π + 2 πn

x = 2 πn, x = π + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for (tan(3x))/(tan(2x))=1 ?
The general solution for (tan(3x))/(tan(2x))=1 is No Solution for x\in\mathbb{R}