What is the general solution for tan(x)=tan(2x-30) ?

The general solution for tan(x)=tan(2x-30) is x=-2pin+30,x=-pi+30-2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

tan(x)=tan(2x-30)

Solution

tan (x) = tan (2 x - 30)

Solution

x = - 2 πn + 30, x = - π + 30 - 2 πn

Degrees

x = 1718.87338 \dots^{\circ} - 36 0^{\circ} n, x = 1538.87338 \dots^{\circ} - 36 0^{\circ} n

Solution steps

tan (x) = tan (2 x - 30)

Subtract

tan (2 x - 30)

from both sides

tan (x) - tan (2 x - 30) = 0

Express with sin, cos

- tan (- 30 + 2 x) + tan (x)

Use the basic trigonometric identity:

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

= - \frac{sin ( - 30 + 2 x )}{cos ( - 30 + 2 x )} + tan (x)

Use the basic trigonometric identity:

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

= - \frac{sin ( - 30 + 2 x )}{cos ( - 30 + 2 x )} + \frac{sin ( x )}{cos ( x )}

Simplify

- \frac{sin ( - 30 + 2 x )}{cos ( - 30 + 2 x )} + \frac{sin ( x )}{cos ( x )} : \frac{- sin ( - 30 + 2 x ) cos ( x ) + sin ( x ) cos ( 2 x - 30 )}{cos ( 2 x - 30 ) cos ( x )}

- \frac{sin ( - 30 + 2 x )}{cos ( - 30 + 2 x )} + \frac{sin ( x )}{cos ( x )}

Least Common Multiplier of

cos (- 30 + 2 x), cos (x) : cos (2 x - 30) cos (x)

cos (- 30 + 2 x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (- 30 + 2 x)

cos (x)

= cos (2 x - 30) cos (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 - 30) cos (x)

For

\frac{sin ( - 30 + 2 x )}{cos ( - 30 + 2 x )} :

multiply the denominator and numerator by

cos (x)

\frac{sin ( - 30 + 2 x )}{cos ( - 30 + 2 x )} = \frac{sin ( - 30 + 2 x ) cos ( x )}{cos ( - 30 + 2 x ) cos ( x )}

For

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

multiply the denominator and numerator by

cos (2 x - 30)

\frac{sin ( x )}{cos ( x )} = \frac{sin ( x ) cos ( 2 x - 30 )}{cos ( x ) cos ( 2 x - 30 )}

= - \frac{sin ( - 30 + 2 x ) cos ( x )}{cos ( - 30 + 2 x ) cos ( x )} + \frac{sin ( x ) cos ( 2 x - 30 )}{cos ( x ) cos ( 2 x - 30 )}

Since the denominators are equal, combine the fractions:

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

= \frac{- sin ( - 30 + 2 x ) cos ( x ) + sin ( x ) cos ( 2 x - 30 )}{cos ( 2 x - 30 ) cos ( x )}

= \frac{- sin ( - 30 + 2 x ) cos ( x ) + sin ( x ) cos ( 2 x - 30 )}{cos ( 2 x - 30 ) cos ( x )}

\frac{cos ( - 30 + 2 x ) sin ( x ) - cos ( x ) sin ( - 30 + 2 x )}{cos ( - 30 + 2 x ) cos ( x )} = 0

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

cos (- 30 + 2 x) sin (x) - cos (x) sin (- 30 + 2 x) = 0

Rewrite using trig identities

cos (- 30 + 2 x) sin (x) - cos (x) sin (- 30 + 2 x)

Use the Angle Difference identity:

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

= sin (x - (- 30 + 2 x))

sin (x - (- 30 + 2 x)) = 0

General solutions for

sin (x - (- 30 + 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}

x - (- 30 + 2 x) = 0 + 2 πn, x - (- 30 + 2 x) = π + 2 πn

x - (- 30 + 2 x) = 0 + 2 πn, x - (- 30 + 2 x) = π + 2 πn

Solve

x - (- 30 + 2 x) = 0 + 2 πn : x = - 2 πn + 30

x - (- 30 + 2 x) = 0 + 2 πn

0 + 2 πn = 2 πn

x - (- 30 + 2 x) = 2 πn

Expand

x - (- 30 + 2 x) : - x + 30

x - (- 30 + 2 x)

- (- 30 + 2 x) : 30 - 2 x

- (- 30 + 2 x)

Distribute parentheses

= - (- 30) - (2 x)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= 30 - 2 x

= x + 30 - 2 x

Simplify

x + 30 - 2 x : - x + 30

x + 30 - 2 x

Group like terms

= x - 2 x + 30

Add similar elements:

x - 2 x = - x

= - x + 30

= - x + 30

- x + 30 = 2 πn

Move

30

to the right side

- x + 30 = 2 πn

Subtract

30

from both sides

- x + 30 - 30 = 2 πn - 30

Simplify

- x = 2 πn - 30

- x = 2 πn - 30

Divide both sides by

- 1

- x = 2 πn - 30

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{2 πn}{- 1} - \frac{30}{- 1}

Simplify

\frac{- x}{- 1} = \frac{2 πn}{- 1} - \frac{30}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{2 πn}{- 1} - \frac{30}{- 1} : - 2 πn + 30

\frac{2 πn}{- 1} - \frac{30}{- 1}

\frac{2 πn}{- 1} = - 2 πn

\frac{2 πn}{- 1}

Apply the fraction rule:

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

= - \frac{2 πn}{1}

Apply rule

\frac{a}{1} = a

= - 2 πn

= - 2 πn - \frac{30}{- 1}

\frac{30}{- 1} = - 30

\frac{30}{- 1}

Apply the fraction rule:

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

= - \frac{30}{1}

Apply rule

\frac{a}{1} = a

= - 30

= - 2 πn - (- 30)

Apply rule

- (- a) = a

= - 2 πn + 30

x = - 2 πn + 30

x = - 2 πn + 30

x = - 2 πn + 30

Solve

x - (- 30 + 2 x) = π + 2 πn : x = - π + 30 - 2 πn

x - (- 30 + 2 x) = π + 2 πn

Expand

x - (- 30 + 2 x) : - x + 30

x - (- 30 + 2 x)

- (- 30 + 2 x) : 30 - 2 x

- (- 30 + 2 x)

Distribute parentheses

= - (- 30) - (2 x)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= 30 - 2 x

= x + 30 - 2 x

Simplify

x + 30 - 2 x : - x + 30

x + 30 - 2 x

Group like terms

= x - 2 x + 30

Add similar elements:

x - 2 x = - x

= - x + 30

= - x + 30

- x + 30 = π + 2 πn

Move

30

to the right side

- x + 30 = π + 2 πn

Subtract

30

from both sides

- x + 30 - 30 = π + 2 πn - 30

Simplify

- x = π + 2 πn - 30

- x = π + 2 πn - 30

Divide both sides by

- 1

- x = π + 2 πn - 30

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{π}{- 1} + \frac{2 πn}{- 1} - \frac{30}{- 1}

Simplify

\frac{- x}{- 1} = \frac{π}{- 1} + \frac{2 πn}{- 1} - \frac{30}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{π}{- 1} + \frac{2 πn}{- 1} - \frac{30}{- 1} : - π + 30 - 2 πn

\frac{π}{- 1} + \frac{2 πn}{- 1} - \frac{30}{- 1}

Group like terms

= \frac{π}{- 1} - \frac{30}{- 1} + \frac{2 πn}{- 1}

\frac{π}{- 1} = - π

\frac{π}{- 1}

Apply the fraction rule:

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

= - \frac{π}{1}

Apply rule

\frac{a}{1} = a

= - π

= - π - \frac{30}{- 1} + \frac{2 πn}{- 1}

\frac{30}{- 1} = - 30

\frac{30}{- 1}

Apply the fraction rule:

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

= - \frac{30}{1}

Apply rule

\frac{a}{1} = a

= - 30

\frac{2 πn}{- 1} = - 2 πn

\frac{2 πn}{- 1}

Apply the fraction rule:

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

= - \frac{2 πn}{1}

Apply rule

\frac{a}{1} = a

= - 2 πn

= - π - (- 30) - 2 πn

Apply rule

- (- a) = a

= - π + 30 - 2 πn

x = - π + 30 - 2 πn

x = - π + 30 - 2 πn

x = - π + 30 - 2 πn

x = - 2 πn + 30, x = - π + 30 - 2 πn

Graph

Popular Examples

sqrt(2)sin(x/2+pi/6)=1,x<4pi,0

16cos^2(x)-9=0

cos(θ)=0.476

sin(x)= 15/19

12cos(x)+6sin(2x)=0,0<= x<= 2pi

Frequently Asked Questions (FAQ)

What is the general solution for tan(x)=tan(2x-30) ?
The general solution for tan(x)=tan(2x-30) is x=-2pin+30,x=-pi+30-2pin