What is the general solution for tan(x+20)=cot(x) ?

The general solution for tan(x+20)=cot(x) is x=180n+45-10,x=180n-10+135

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

tan(x+20)=cot(x)

Solution

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

Solution

x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}, x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

Radians

x = \frac{π}{4} - \frac{π}{18} + πn, x = - \frac{π}{18} + \frac{3 π}{4} + πn

Solution steps

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

Subtract

cot (x)

from both sides

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Least Common Multiplier of

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

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

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

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

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

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

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

For

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

multiply the denominator and numerator by

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

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

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

sin (x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Rewrite using trig identities

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

Use the Angle Sum identity:

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

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

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

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

Divide both sides by

- 1

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

Divide both sides by

- 1

\frac{- cos ( \frac{18 0 ^{\circ} + 9 x}{9} + x )}{- 1} = \frac{0}{- 1}

Simplify

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

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

General solutions for

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

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

\frac{18 0 ^{\circ} + 9 x}{9} + x = 9 0^{\circ} + 36 0^{\circ} n, \frac{18 0 ^{\circ} + 9 x}{9} + x = 27 0^{\circ} + 36 0^{\circ} n

\frac{18 0 ^{\circ} + 9 x}{9} + x = 9 0^{\circ} + 36 0^{\circ} n, \frac{18 0 ^{\circ} + 9 x}{9} + x = 27 0^{\circ} + 36 0^{\circ} n

Solve

\frac{18 0 ^{\circ} + 9 x}{9} + x = 9 0^{\circ} + 36 0^{\circ} n : x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

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

Multiply both sides by

9

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

Multiply both sides by

9

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 + x \cdot 9 = 9 0^{\circ} \cdot 9 + 36 0^{\circ} n \cdot 9

Simplify

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

x \cdot 9 : 9 x

x \cdot 9

Apply the commutative law:

x \cdot 9 = 9 x

9 x

Simplify

9 0^{\circ} \cdot 9 : 81 0^{\circ}

9 0^{\circ} \cdot 9

Multiply fractions:

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

= 81 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 x = 81 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} + 18 x = 81 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} + 18 x = 81 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} + 18 x = 81 0^{\circ} + 324 0^{\circ} n

Move

18 0^{\circ}

to the right side

18 0^{\circ} + 18 x = 81 0^{\circ} + 324 0^{\circ} n

Subtract

18 0^{\circ}

from both sides

18 0^{\circ} + 18 x - 18 0^{\circ} = 81 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

Simplify

18 x = 81 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

18 x = 81 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

Divide both sides by

18

18 x = 81 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

Divide both sides by

18

\frac{18 x}{18} = \frac{81 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ}

Simplify

\frac{18 x}{18} = \frac{81 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ}

Simplify

\frac{18 x}{18} : x

\frac{18 x}{18}

Divide the numbers:

\frac{18}{18} = 1

= x

Simplify

\frac{81 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ} : 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

\frac{81 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ}

Group like terms

= - 1 0^{\circ} + \frac{324 0 ^{\circ} n}{18} + \frac{81 0 ^{\circ}}{18}

\frac{324 0 ^{\circ} n}{18} = 18 0^{\circ} n

\frac{324 0 ^{\circ} n}{18}

Divide the numbers:

\frac{18}{18} = 1

= 18 0^{\circ} n

\frac{81 0 ^{\circ}}{18} = 4 5^{\circ}

\frac{81 0 ^{\circ}}{18}

Apply the fraction rule:

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

= \frac{162 0 ^{\circ}}{2 \cdot 18}

Multiply the numbers:

2 \cdot 18 = 36

= 4 5^{\circ}

Cancel the common factor:

9

= 4 5^{\circ}

= - 1 0^{\circ} + 18 0^{\circ} n + 4 5^{\circ}

Group like terms

= 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

Solve

\frac{18 0 ^{\circ} + 9 x}{9} + x = 27 0^{\circ} + 36 0^{\circ} n : x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

\frac{18 0 ^{\circ} + 9 x}{9} + x = 27 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

9

\frac{18 0 ^{\circ} + 9 x}{9} + x = 27 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

9

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 + x \cdot 9 = 27 0^{\circ} \cdot 9 + 36 0^{\circ} n \cdot 9

Simplify

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 + x \cdot 9 = 27 0^{\circ} \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

x \cdot 9 : 9 x

x \cdot 9

Apply the commutative law:

x \cdot 9 = 9 x

9 x

Simplify

27 0^{\circ} \cdot 9 : 243 0^{\circ}

27 0^{\circ} \cdot 9

Multiply fractions:

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

= 243 0^{\circ}

Multiply the numbers:

3 \cdot 9 = 27

= 243 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 x = 243 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} + 18 x = 243 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} + 18 x = 243 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} + 18 x = 243 0^{\circ} + 324 0^{\circ} n

Move

18 0^{\circ}

to the right side

18 0^{\circ} + 18 x = 243 0^{\circ} + 324 0^{\circ} n

Subtract

18 0^{\circ}

from both sides

18 0^{\circ} + 18 x - 18 0^{\circ} = 243 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

Simplify

18 x = 243 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

18 x = 243 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

Divide both sides by

18

18 x = 243 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

Divide both sides by

18

\frac{18 x}{18} = \frac{243 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ}

Simplify

\frac{18 x}{18} = \frac{243 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ}

Simplify

\frac{18 x}{18} : x

\frac{18 x}{18}

Divide the numbers:

\frac{18}{18} = 1

= x

Simplify

\frac{243 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ} : 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

\frac{243 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ}

Group like terms

= - 1 0^{\circ} + \frac{324 0 ^{\circ} n}{18} + \frac{243 0 ^{\circ}}{18}

\frac{324 0 ^{\circ} n}{18} = 18 0^{\circ} n

\frac{324 0 ^{\circ} n}{18}

Divide the numbers:

\frac{18}{18} = 1

= 18 0^{\circ} n

\frac{243 0 ^{\circ}}{18} = 13 5^{\circ}

\frac{243 0 ^{\circ}}{18}

Apply the fraction rule:

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

= \frac{486 0 ^{\circ}}{2 \cdot 18}

Multiply the numbers:

2 \cdot 18 = 36

= 13 5^{\circ}

Cancel the common factor:

9

= 13 5^{\circ}

= - 1 0^{\circ} + 18 0^{\circ} n + 13 5^{\circ}

Group like terms

= 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}, x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

Graph

Popular Examples

3csc^2(x)+1=5

6cos^2(x)-12cos(x)+6=0

6sin^2(θ)-10sin(θ)-8=-3sin(θ)-3

sin(4θ)-sin(6θ)=0

sin(x)= 3/10

Frequently Asked Questions (FAQ)

What is the general solution for tan(x+20)=cot(x) ?
The general solution for tan(x+20)=cot(x) is x=180n+45-10,x=180n-10+135