What is the general solution for tan(x-20)=tan(2x+10) ?

The general solution for tan(x-20)=tan(2x+10) is x=-360n-20-10,x=-10-200-360n

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tan(x-20)=tan(2x+10)

Solution

tan (x - 2 0^{\circ}) = tan (2 x + 10)

Solution

x = - 36 0^{\circ} n - 2 0^{\circ} - 10, x = - 10 - 20 0^{\circ} - 36 0^{\circ} n

Radians

x = - \frac{π}{9} - 10 - 2 πn, x = - 10 - \frac{10 π}{9} - 2 πn

Solution steps

tan (x - 2 0^{\circ}) = tan (2 x + 10)

Subtract

tan (2 x + 10)

from both sides

tan (x - 2 0^{\circ}) - tan (2 x + 10) = 0

Express with sin, cos

tan (- 2 0^{\circ} + x) - tan (10 + 2 x)

Use the basic trigonometric identity:

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

= \frac{sin ( - 2 0 ^{\circ} + x )}{cos ( - 2 0 ^{\circ} + x )} - tan (10 + 2 x)

Use the basic trigonometric identity:

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

= \frac{sin ( - 2 0 ^{\circ} + x )}{cos ( - 2 0 ^{\circ} + x )} - \frac{sin ( 10 + 2 x )}{cos ( 10 + 2 x )}

Simplify

\frac{sin ( - 2 0 ^{\circ} + x )}{cos ( - 2 0 ^{\circ} + x )} - \frac{sin ( 10 + 2 x )}{cos ( 10 + 2 x )} : \frac{sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( 2 x + 10 ) - sin ( 10 + 2 x ) cos ( \frac{9 x - 18 0 ^{\circ}}{9} )}{cos ( \frac{9 x - 18 0 ^{\circ}}{9} ) cos ( 2 x + 10 )}

\frac{sin ( - 2 0 ^{\circ} + x )}{cos ( - 2 0 ^{\circ} + x )} - \frac{sin ( 10 + 2 x )}{cos ( 10 + 2 x )}

\frac{sin ( - 2 0 ^{\circ} + x )}{cos ( - 2 0 ^{\circ} + x )} = \frac{sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} )}

\frac{sin ( - 2 0 ^{\circ} + x )}{cos ( - 2 0 ^{\circ} + x )}

Join

- 2 0^{\circ} + x : \frac{- 18 0 ^{\circ} + 9 x}{9}

- 2 0^{\circ} + x

Convert element to fraction:

x = \frac{x 9}{9}

= - 2 0^{\circ} + \frac{x \cdot 9}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{- 18 0 ^{\circ} + x \cdot 9}{9}

= \frac{sin ( - 2 0 ^{\circ} + x )}{cos ( \frac{- 18 0 ^{\circ} + x \cdot 9}{9} )}

Join

- 2 0^{\circ} + x : \frac{- 18 0 ^{\circ} + 9 x}{9}

- 2 0^{\circ} + x

Convert element to fraction:

x = \frac{x 9}{9}

= - 2 0^{\circ} + \frac{x \cdot 9}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{- 18 0 ^{\circ} + x \cdot 9}{9}

= \frac{sin ( \frac{- 18 0 ^{\circ} + x \cdot 9}{9} )}{cos ( \frac{- 18 0 ^{\circ} + x \cdot 9}{9} )}

= \frac{sin ( \frac{9 x - 18 0 ^{\circ}}{9} )}{cos ( \frac{9 x - 18 0 ^{\circ}}{9} )} - \frac{sin ( 2 x + 10 )}{cos ( 2 x + 10 )}

Least Common Multiplier of

cos (\frac{- 18 0 ^{\circ} + x 9}{9}), cos (10 + 2 x) : cos (\frac{9 x - 18 0 ^{\circ}}{9}) cos (2 x + 10)

cos (\frac{- 18 0 ^{\circ} + x \cdot 9}{9}), cos (10 + 2 x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (\frac{- 18 0 ^{\circ} + x 9}{9})

cos (10 + 2 x)

= cos (\frac{9 x - 18 0 ^{\circ}}{9}) cos (2 x + 10)

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 (\frac{9 x - 18 0 ^{\circ}}{9}) cos (2 x + 10)

For

\frac{sin ( \frac{- 18 0 ^{\circ} + x \cdot 9}{9} )}{cos ( \frac{- 18 0 ^{\circ} + x \cdot 9}{9} )} :

multiply the denominator and numerator by

cos (2 x + 10)

\frac{sin ( \frac{- 18 0 ^{\circ} + x \cdot 9}{9} )}{cos ( \frac{- 18 0 ^{\circ} + x \cdot 9}{9} )} = \frac{sin ( \frac{- 18 0 ^{\circ} + x \cdot 9}{9} ) cos ( 2 x + 10 )}{cos ( \frac{- 18 0 ^{\circ} + x \cdot 9}{9} ) cos ( 2 x + 10 )}

For

\frac{sin ( 10 + 2 x )}{cos ( 10 + 2 x )} :

multiply the denominator and numerator by

cos (\frac{9 x - 18 0 ^{\circ}}{9})

\frac{sin ( 10 + 2 x )}{cos ( 10 + 2 x )} = \frac{sin ( 10 + 2 x ) cos ( \frac{9 x - 18 0 ^{\circ}}{9} )}{cos ( 10 + 2 x ) cos ( \frac{9 x - 18 0 ^{\circ}}{9} )}

= \frac{sin ( \frac{- 18 0 ^{\circ} + x \cdot 9}{9} ) cos ( 2 x + 10 )}{cos ( \frac{- 18 0 ^{\circ} + x \cdot 9}{9} ) cos ( 2 x + 10 )} - \frac{sin ( 10 + 2 x ) cos ( \frac{9 x - 18 0 ^{\circ}}{9} )}{cos ( 10 + 2 x ) cos ( \frac{9 x - 18 0 ^{\circ}}{9} )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( \frac{- 18 0 ^{\circ} + x \cdot 9}{9} ) cos ( 2 x + 10 ) - sin ( 10 + 2 x ) cos ( \frac{9 x - 18 0 ^{\circ}}{9} )}{cos ( \frac{9 x - 18 0 ^{\circ}}{9} ) cos ( 2 x + 10 )}

= \frac{sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( 2 x + 10 ) - sin ( 10 + 2 x ) cos ( \frac{9 x - 18 0 ^{\circ}}{9} )}{cos ( \frac{9 x - 18 0 ^{\circ}}{9} ) cos ( 2 x + 10 )}

\frac{cos ( 10 + 2 x ) sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) - cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) sin ( 10 + 2 x )}{cos ( 10 + 2 x ) cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} )} = 0

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

cos (10 + 2 x) sin (\frac{- 18 0 ^{\circ} + 9 x}{9}) - cos (\frac{- 18 0 ^{\circ} + 9 x}{9}) sin (10 + 2 x) = 0

Rewrite using trig identities

cos (10 + 2 x) sin (\frac{- 18 0 ^{\circ} + 9 x}{9}) - cos (\frac{- 18 0 ^{\circ} + 9 x}{9}) sin (10 + 2 x)

Use the Angle Difference identity:

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

= sin (\frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x))

sin (\frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x)) = 0

General solutions for

sin (\frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x)) = 0

sin (x)

periodicity table with

36 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x) = 0 + 36 0^{\circ} n, \frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x) = 18 0^{\circ} + 36 0^{\circ} n

\frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x) = 0 + 36 0^{\circ} n, \frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x) = 18 0^{\circ} + 36 0^{\circ} n

Solve

\frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x) = 0 + 36 0^{\circ} n : x = - 36 0^{\circ} n - 2 0^{\circ} - 10

\frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x) = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

\frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x) = 36 0^{\circ} n

Multiply both sides by

9

\frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x) = 36 0^{\circ} n

Multiply both sides by

9

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9 - (10 + 2 x) \cdot 9 = 36 0^{\circ} n \cdot 9

Simplify

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9 - (10 + 2 x) \cdot 9 = 36 0^{\circ} n \cdot 9

Simplify

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9 : - 18 0^{\circ} + 9 x

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9

Multiply fractions:

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

= \frac{( - 18 0 ^{\circ} + 9 x ) \cdot 9}{9}

Cancel the common factor:

9

= - - 18 0^{\circ} + 9 x

Simplify

(10 + 2 x) \cdot 9 : 9 (10 + 2 x)

(10 + 2 x) \cdot 9

Apply the commutative law:

(10 + 2 x) \cdot 9 = 9 (10 + 2 x)

9 (10 + 2 x)

Simplify

36 0^{\circ} n \cdot 9 : 324 0^{\circ} n

36 0^{\circ} n \cdot 9

Multiply the numbers:

2 \cdot 9 = 18

= 324 0^{\circ} n

- 18 0^{\circ} + 9 x - 9 (10 + 2 x) = 324 0^{\circ} n

- 18 0^{\circ} + 9 x - 9 (10 + 2 x) = 324 0^{\circ} n

- 18 0^{\circ} + 9 x - 9 (10 + 2 x) = 324 0^{\circ} n

Expand

- 18 0^{\circ} + 9 x - 9 (10 + 2 x) : - 9 x - 18 0^{\circ} - 90

- 18 0^{\circ} + 9 x - 9 (10 + 2 x)

Expand

- 9 (10 + 2 x) : - 90 - 18 x

- 9 (10 + 2 x)

Apply the distributive law:

a (b + c) = ab + a c

a = - 9, b = 10, c = 2 x

= - 9 \cdot 10 + (- 9) \cdot 2 x

Apply minus-plus rules

+ (- a) = - a

= - 9 \cdot 10 - 9 \cdot 2 x

Simplify

- 9 \cdot 10 - 9 \cdot 2 x : - 90 - 18 x

- 9 \cdot 10 - 9 \cdot 2 x

Multiply the numbers:

9 \cdot 10 = 90

= - 90 - 9 \cdot 2 x

Multiply the numbers:

9 \cdot 2 = 18

= - 90 - 18 x

= - 90 - 18 x

= - 18 0^{\circ} + 9 x - 90 - 18 x

Simplify

- 18 0^{\circ} + 9 x - 90 - 18 x : - 9 x - 18 0^{\circ} - 90

- 18 0^{\circ} + 9 x - 90 - 18 x

Group like terms

= 9 x - 18 x - 18 0^{\circ} - 90

Add similar elements:

9 x - 18 x = - 9 x

= - 9 x - 18 0^{\circ} - 90

= - 9 x - 18 0^{\circ} - 90

- 9 x - 18 0^{\circ} - 90 = 324 0^{\circ} n

Move

18 0^{\circ}

to the right side

- 9 x - 18 0^{\circ} - 90 = 324 0^{\circ} n

Add

18 0^{\circ}

to both sides

- 9 x - 18 0^{\circ} - 90 + 18 0^{\circ} = 324 0^{\circ} n + 18 0^{\circ}

Simplify

- 9 x - 90 = 324 0^{\circ} n + 18 0^{\circ}

- 9 x - 90 = 324 0^{\circ} n + 18 0^{\circ}

Move

90

to the right side

- 9 x - 90 = 324 0^{\circ} n + 18 0^{\circ}

Add

90

to both sides

- 9 x - 90 + 90 = 324 0^{\circ} n + 18 0^{\circ} + 90

Simplify

- 9 x = 324 0^{\circ} n + 18 0^{\circ} + 90

- 9 x = 324 0^{\circ} n + 18 0^{\circ} + 90

Divide both sides by

- 9

- 9 x = 324 0^{\circ} n + 18 0^{\circ} + 90

Divide both sides by

- 9

\frac{- 9 x}{- 9} = \frac{324 0 ^{\circ} n}{- 9} + \frac{18 0 ^{\circ}}{- 9} + \frac{90}{- 9}

Simplify

\frac{- 9 x}{- 9} = \frac{324 0 ^{\circ} n}{- 9} + \frac{18 0 ^{\circ}}{- 9} + \frac{90}{- 9}

Simplify

\frac{- 9 x}{- 9} : x

\frac{- 9 x}{- 9}

Apply the fraction rule:

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

= \frac{9 x}{9}

Divide the numbers:

\frac{9}{9} = 1

= x

Simplify

\frac{324 0 ^{\circ} n}{- 9} + \frac{18 0 ^{\circ}}{- 9} + \frac{90}{- 9} : - 36 0^{\circ} n - 2 0^{\circ} - 10

\frac{324 0 ^{\circ} n}{- 9} + \frac{18 0 ^{\circ}}{- 9} + \frac{90}{- 9}

\frac{324 0 ^{\circ} n}{- 9} = - 36 0^{\circ} n

\frac{324 0 ^{\circ} n}{- 9}

Apply the fraction rule:

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

= - \frac{324 0 ^{\circ} n}{9}

Divide the numbers:

\frac{18}{9} = 2

= - 36 0^{\circ} n

= - 36 0^{\circ} n + \frac{18 0 ^{\circ}}{- 9} + \frac{90}{- 9}

Apply the fraction rule:

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

= - 36 0^{\circ} n - 2 0^{\circ} + \frac{90}{- 9}

\frac{90}{- 9} = - 10

\frac{90}{- 9}

Apply the fraction rule:

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

= - \frac{90}{9}

Divide the numbers:

\frac{90}{9} = 10

= - 10

= - 36 0^{\circ} n - 2 0^{\circ} - 10

x = - 36 0^{\circ} n - 2 0^{\circ} - 10

x = - 36 0^{\circ} n - 2 0^{\circ} - 10

x = - 36 0^{\circ} n - 2 0^{\circ} - 10

Solve

\frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x) = 18 0^{\circ} + 36 0^{\circ} n : x = - 10 - 20 0^{\circ} - 36 0^{\circ} n

\frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x) = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

9

\frac{- 18 0 ^{\circ} + 9 x}{9} - (10 + 2 x) = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

9

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9 - (10 + 2 x) \cdot 9 = 18 0^{\circ} 9 + 36 0^{\circ} n \cdot 9

Simplify

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9 - (10 + 2 x) \cdot 9 = 18 0^{\circ} 9 + 36 0^{\circ} n \cdot 9

Simplify

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9 : - 18 0^{\circ} + 9 x

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9

Multiply fractions:

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

= \frac{( - 18 0 ^{\circ} + 9 x ) \cdot 9}{9}

Cancel the common factor:

9

= - - 18 0^{\circ} + 9 x

Simplify

(10 + 2 x) \cdot 9 : 9 (10 + 2 x)

(10 + 2 x) \cdot 9

Apply the commutative law:

(10 + 2 x) \cdot 9 = 9 (10 + 2 x)

9 (10 + 2 x)

Simplify

18 0^{\circ} 9 : 162 0^{\circ}

18 0^{\circ} 9

Apply the commutative law:

18 0^{\circ} 9 = 162 0^{\circ}

162 0^{\circ}

Simplify

36 0^{\circ} n \cdot 9 : 324 0^{\circ} n

36 0^{\circ} n \cdot 9

Multiply the numbers:

2 \cdot 9 = 18

= 324 0^{\circ} n

- 18 0^{\circ} + 9 x - 9 (10 + 2 x) = 162 0^{\circ} + 324 0^{\circ} n

- 18 0^{\circ} + 9 x - 9 (10 + 2 x) = 162 0^{\circ} + 324 0^{\circ} n

- 18 0^{\circ} + 9 x - 9 (10 + 2 x) = 162 0^{\circ} + 324 0^{\circ} n

Expand

- 18 0^{\circ} + 9 x - 9 (10 + 2 x) : - 9 x - 18 0^{\circ} - 90

- 18 0^{\circ} + 9 x - 9 (10 + 2 x)

Expand

- 9 (10 + 2 x) : - 90 - 18 x

- 9 (10 + 2 x)

Apply the distributive law:

a (b + c) = ab + a c

a = - 9, b = 10, c = 2 x

= - 9 \cdot 10 + (- 9) \cdot 2 x

Apply minus-plus rules

+ (- a) = - a

= - 9 \cdot 10 - 9 \cdot 2 x

Simplify

- 9 \cdot 10 - 9 \cdot 2 x : - 90 - 18 x

- 9 \cdot 10 - 9 \cdot 2 x

Multiply the numbers:

9 \cdot 10 = 90

= - 90 - 9 \cdot 2 x

Multiply the numbers:

9 \cdot 2 = 18

= - 90 - 18 x

= - 90 - 18 x

= - 18 0^{\circ} + 9 x - 90 - 18 x

Simplify

- 18 0^{\circ} + 9 x - 90 - 18 x : - 9 x - 18 0^{\circ} - 90

- 18 0^{\circ} + 9 x - 90 - 18 x

Group like terms

= 9 x - 18 x - 18 0^{\circ} - 90

Add similar elements:

9 x - 18 x = - 9 x

= - 9 x - 18 0^{\circ} - 90

= - 9 x - 18 0^{\circ} - 90

- 9 x - 18 0^{\circ} - 90 = 162 0^{\circ} + 324 0^{\circ} n

Move

18 0^{\circ}

to the right side

- 9 x - 18 0^{\circ} - 90 = 162 0^{\circ} + 324 0^{\circ} n

Add

18 0^{\circ}

to both sides

- 9 x - 18 0^{\circ} - 90 + 18 0^{\circ} = 162 0^{\circ} + 324 0^{\circ} n + 18 0^{\circ}

Simplify

- 9 x - 90 = 180 0^{\circ} + 324 0^{\circ} n

- 9 x - 90 = 180 0^{\circ} + 324 0^{\circ} n

Move

90

to the right side

- 9 x - 90 = 180 0^{\circ} + 324 0^{\circ} n

Add

90

to both sides

- 9 x - 90 + 90 = 180 0^{\circ} + 324 0^{\circ} n + 90

Simplify

- 9 x = 180 0^{\circ} + 324 0^{\circ} n + 90

- 9 x = 180 0^{\circ} + 324 0^{\circ} n + 90

Divide both sides by

- 9

- 9 x = 180 0^{\circ} + 324 0^{\circ} n + 90

Divide both sides by

- 9

\frac{- 9 x}{- 9} = \frac{180 0 ^{\circ}}{- 9} + \frac{324 0 ^{\circ} n}{- 9} + \frac{90}{- 9}

Simplify

\frac{- 9 x}{- 9} = \frac{180 0 ^{\circ}}{- 9} + \frac{324 0 ^{\circ} n}{- 9} + \frac{90}{- 9}

Simplify

\frac{- 9 x}{- 9} : x

\frac{- 9 x}{- 9}

Apply the fraction rule:

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

= \frac{9 x}{9}

Divide the numbers:

\frac{9}{9} = 1

= x

Simplify

\frac{180 0 ^{\circ}}{- 9} + \frac{324 0 ^{\circ} n}{- 9} + \frac{90}{- 9} : - 10 - 20 0^{\circ} - 36 0^{\circ} n

\frac{180 0 ^{\circ}}{- 9} + \frac{324 0 ^{\circ} n}{- 9} + \frac{90}{- 9}

Group like terms

= \frac{90}{- 9} + \frac{180 0 ^{\circ}}{- 9} + \frac{324 0 ^{\circ} n}{- 9}

\frac{90}{- 9} = - 10

\frac{90}{- 9}

Apply the fraction rule:

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

= - \frac{90}{9}

Divide the numbers:

\frac{90}{9} = 10

= - 10

= - 10 + \frac{180 0 ^{\circ}}{- 9} + \frac{324 0 ^{\circ} n}{- 9}

Apply the fraction rule:

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

= - 10 - 20 0^{\circ} + \frac{324 0 ^{\circ} n}{- 9}

\frac{324 0 ^{\circ} n}{- 9} = - 36 0^{\circ} n

\frac{324 0 ^{\circ} n}{- 9}

Apply the fraction rule:

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

= - \frac{324 0 ^{\circ} n}{9}

Divide the numbers:

\frac{18}{9} = 2

= - 36 0^{\circ} n

= - 10 - 20 0^{\circ} - 36 0^{\circ} n

x = - 10 - 20 0^{\circ} - 36 0^{\circ} n

x = - 10 - 20 0^{\circ} - 36 0^{\circ} n

x = - 10 - 20 0^{\circ} - 36 0^{\circ} n

x = - 36 0^{\circ} n - 2 0^{\circ} - 10, x = - 10 - 20 0^{\circ} - 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x-20)=tan(2x+10) ?
The general solution for tan(x-20)=tan(2x+10) is x=-360n-20-10,x=-10-200-360n