What is the general solution for tan(3x)=tan(pix) ?

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

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tan(3x)=tan(pix)

Solution

tan (3 x) = tan (π x)

Solution

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

Degrees

x = 0^{\circ} - 2542.50479 \dots^{\circ} n, x = 0^{\circ} - 3813.75718 \dots^{\circ} n

Solution steps

tan (3 x) = tan (π x)

Subtract

tan (π x)

from both sides

tan (3 x) - tan (π x) = 0

Express with sin, cos

tan (3 x) - tan (x π)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

\frac{sin ( 3 x )}{cos ( 3 x )} - \frac{sin ( x π )}{cos ( x π )}

Least Common Multiplier of

cos (3 x), cos (x π) : cos (3 x) cos (π x)

cos (3 x), cos (x π)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (3 x)

cos (x π)

= cos (3 x) cos (π 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 (3 x) cos (π x)

For

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

multiply the denominator and numerator by

cos (π x)

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

For

\frac{sin ( x π )}{cos ( x π )} :

multiply the denominator and numerator by

cos (3 x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

- cos (3 x) sin (x π) + cos (x π) sin (3 x) = 0

Rewrite using trig identities

- cos (3 x) sin (x π) + cos (x π) sin (3 x)

Use the Angle Difference identity:

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

= sin (3 x - x π)

sin (3 x - x π) = 0

General solutions for

sin (3 x - x π) = 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 - x π = 0 + 2 πn, 3 x - x π = π + 2 πn

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

Solve

3 x - x π = 0 + 2 πn : x = \frac{2 πn}{3 - π}

3 x - x π = 0 + 2 πn

0 + 2 πn = 2 πn

3 x - x π = 2 πn

Factor

3 x - x π : (3 - π) x

3 x - x π

Factor out common term

x

= x (3 - π)

(3 - π) x = 2 πn

Divide both sides by

3 - π

(3 - π) x = 2 πn

Divide both sides by

3 - π

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

Simplify

x = \frac{2 πn}{3 - π}

x = \frac{2 πn}{3 - π}

Solve

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

3 x - x π = π + 2 πn

Factor

3 x - x π : (3 - π) x

3 x - x π

Factor out common term

x

= x (3 - π)

(3 - π) x = π + 2 πn

Divide both sides by

3 - π

(3 - π) x = π + 2 πn

Divide both sides by

3 - π

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

Simplify

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

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

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

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Frequently Asked Questions (FAQ)

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