What is the general solution for tan(β+10)=cot(2β-10) ?

The general solution for tan(β+10)=cot(2β-10) is β=30+(360n)/3 ,β=90+(360n)/3

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tan(β+10)=cot(2β-10)

Solution

tan (β + 1 0^{\circ}) = cot (2 β - 1 0^{\circ})

Solution

β = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}, β = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

Radians

β = \frac{π}{6} + \frac{2 π}{3} n, β = \frac{π}{2} + \frac{2 π}{3} n

Solution steps

tan (β + 1 0^{\circ}) = cot (2 β - 1 0^{\circ})

Subtract

cot (2 β - 1 0^{\circ})

from both sides

tan (β + 1 0^{\circ}) - cot (2 β - 1 0^{\circ}) = 0

Simplify

tan (β + 1 0^{\circ}) - cot (2 β - 1 0^{\circ}) : tan (\frac{18 β + 18 0 ^{\circ}}{18}) - cot (\frac{36 β - 18 0 ^{\circ}}{18})

tan (β + 1 0^{\circ}) - cot (2 β - 1 0^{\circ})

Join

β + 1 0^{\circ} : \frac{18 β + 18 0 ^{\circ}}{18}

β + 1 0^{\circ}

Convert element to fraction:

β = \frac{β 18}{18}

= \frac{β \cdot 18}{18} + 1 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{β \cdot 18 + 18 0 ^{\circ}}{18}

= tan (\frac{18 β + 18 0 ^{\circ}}{18}) - cot (2 β - 1 0^{\circ})

Join

2 β - 1 0^{\circ} : \frac{36 β - 18 0 ^{\circ}}{18}

2 β - 1 0^{\circ}

Convert element to fraction:

2 β = \frac{2 β 18}{18}

= \frac{2 β \cdot 18}{18} - 1 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{2 β \cdot 18 - 18 0 ^{\circ}}{18}

Multiply the numbers:

2 \cdot 18 = 36

= \frac{36 β - 18 0 ^{\circ}}{18}

= tan (\frac{18 β + 18 0 ^{\circ}}{18}) - cot (\frac{36 β - 18 0 ^{\circ}}{18})

tan (\frac{18 β + 18 0 ^{\circ}}{18}) - cot (\frac{36 β - 18 0 ^{\circ}}{18}) = 0

Express with sin, cos

- cot (\frac{- 18 0 ^{\circ} + 36 β}{18}) + tan (\frac{18 0 ^{\circ} + 18 β}{18})

Use the basic trigonometric identity:

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

= - \frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} + tan (\frac{18 0 ^{\circ} + 18 β}{18})

Use the basic trigonometric identity:

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

= - \frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} + \frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} )}

Simplify

- \frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} + \frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} )} : \frac{- cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} ) + sin ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}{sin ( \frac{36 β - 18 0 ^{\circ}}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )}

- \frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} + \frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} )}

Least Common Multiplier of

sin (\frac{- 18 0 ^{\circ} + 36 β}{18}), cos (\frac{18 0 ^{\circ} + 18 β}{18}) : sin (\frac{36 β - 18 0 ^{\circ}}{18}) cos (\frac{18 β + 18 0 ^{\circ}}{18})

sin (\frac{- 18 0 ^{\circ} + 36 β}{18}), cos (\frac{18 0 ^{\circ} + 18 β}{18})

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (\frac{- 18 0 ^{\circ} + 36 β}{18})

cos (\frac{18 0 ^{\circ} + 18 β}{18})

= sin (\frac{36 β - 18 0 ^{\circ}}{18}) cos (\frac{18 β + 18 0 ^{\circ}}{18})

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 (\frac{36 β - 18 0 ^{\circ}}{18}) cos (\frac{18 β + 18 0 ^{\circ}}{18})

For

\frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} :

multiply the denominator and numerator by

cos (\frac{18 β + 18 0 ^{\circ}}{18})

\frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} = \frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )}

For

\frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} )} :

multiply the denominator and numerator by

sin (\frac{36 β - 18 0 ^{\circ}}{18})

\frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} )} = \frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}

= - \frac{cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )}{sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )} + \frac{sin ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}

Since the denominators are equal, combine the fractions:

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

= \frac{- cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} ) + sin ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}{sin ( \frac{36 β - 18 0 ^{\circ}}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )}

= \frac{- cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} ) + sin ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{36 β - 18 0 ^{\circ}}{18} )}{sin ( \frac{36 β - 18 0 ^{\circ}}{18} ) cos ( \frac{18 β + 18 0 ^{\circ}}{18} )}

\frac{- cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) cos ( \frac{18 0 ^{\circ} + 18 β}{18} ) + sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} ) sin ( \frac{18 0 ^{\circ} + 18 β}{18} )}{cos ( \frac{18 0 ^{\circ} + 18 β}{18} ) sin ( \frac{- 18 0 ^{\circ} + 36 β}{18} )} = 0

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

- cos (\frac{- 18 0 ^{\circ} + 36 β}{18}) cos (\frac{18 0 ^{\circ} + 18 β}{18}) + sin (\frac{- 18 0 ^{\circ} + 36 β}{18}) sin (\frac{18 0 ^{\circ} + 18 β}{18}) = 0

Rewrite using trig identities

- cos (\frac{- 18 0 ^{\circ} + 36 β}{18}) cos (\frac{18 0 ^{\circ} + 18 β}{18}) + sin (\frac{- 18 0 ^{\circ} + 36 β}{18}) sin (\frac{18 0 ^{\circ} + 18 β}{18})

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} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18})

- cos (\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18}) = 0

Divide both sides by

- 1

- cos (\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18}) = 0

Divide both sides by

- 1

\frac{- cos ( \frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} )}{- 1} = \frac{0}{- 1}

Simplify

cos (\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18}) = 0

cos (\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18}) = 0

General solutions for

cos (\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18}) = 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} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 9 0^{\circ} + 36 0^{\circ} n, \frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 27 0^{\circ} + 36 0^{\circ} n

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

Solve

\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 9 0^{\circ} + 36 0^{\circ} n : β = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

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

Multiply both sides by

18

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

Multiply both sides by

18

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

Simplify

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

Simplify

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18 : - 18 0^{\circ} + 36 β

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18

Multiply fractions:

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

= \frac{( - 18 0 ^{\circ} + 36 β ) \cdot 18}{18}

Cancel the common factor:

18

= - - 18 0^{\circ} + 36 β

Simplify

\frac{18 0 ^{\circ} + 18 β}{18} \cdot 18 : 18 0^{\circ} + 18 β

\frac{18 0 ^{\circ} + 18 β}{18} \cdot 18

Multiply fractions:

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

= \frac{( 18 0 ^{\circ} + 18 β ) \cdot 18}{18}

Cancel the common factor:

18

= 18 0^{\circ} + 18 β

Simplify

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

9 0^{\circ} \cdot 18

Multiply fractions:

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

= 162 0^{\circ}

Divide the numbers:

\frac{18}{2} = 9

= 162 0^{\circ}

Simplify

36 0^{\circ} n \cdot 18 : 648 0^{\circ} n

36 0^{\circ} n \cdot 18

Multiply the numbers:

2 \cdot 18 = 36

= 648 0^{\circ} n

- 18 0^{\circ} + 36 β + 18 0^{\circ} + 18 β = 162 0^{\circ} + 648 0^{\circ} n

54 β = 162 0^{\circ} + 648 0^{\circ} n

54 β = 162 0^{\circ} + 648 0^{\circ} n

54 β = 162 0^{\circ} + 648 0^{\circ} n

Divide both sides by

54

54 β = 162 0^{\circ} + 648 0^{\circ} n

Divide both sides by

54

\frac{54 β}{54} = 3 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Simplify

\frac{54 β}{54} = 3 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Simplify

\frac{54 β}{54} : β

\frac{54 β}{54}

Divide the numbers:

\frac{54}{54} = 1

= β

Simplify

3 0^{\circ} + \frac{648 0 ^{\circ} n}{54} : 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

3 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Cancel

3 0^{\circ} : 3 0^{\circ}

3 0^{\circ}

Cancel the common factor:

9

= 3 0^{\circ}

= 3 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Cancel

\frac{648 0 ^{\circ} n}{54} : \frac{36 0 ^{\circ} n}{3}

\frac{648 0 ^{\circ} n}{54}

Cancel the common factor:

18

= \frac{36 0 ^{\circ} n}{3}

= 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

Solve

\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 27 0^{\circ} + 36 0^{\circ} n : β = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 27 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

18

\frac{- 18 0 ^{\circ} + 36 β}{18} + \frac{18 0 ^{\circ} + 18 β}{18} = 27 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

18

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18 + \frac{18 0 ^{\circ} + 18 β}{18} \cdot 18 = 27 0^{\circ} \cdot 18 + 36 0^{\circ} n \cdot 18

Simplify

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18 + \frac{18 0 ^{\circ} + 18 β}{18} \cdot 18 = 27 0^{\circ} \cdot 18 + 36 0^{\circ} n \cdot 18

Simplify

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18 : - 18 0^{\circ} + 36 β

\frac{- 18 0 ^{\circ} + 36 β}{18} \cdot 18

Multiply fractions:

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

= \frac{( - 18 0 ^{\circ} + 36 β ) \cdot 18}{18}

Cancel the common factor:

18

= - - 18 0^{\circ} + 36 β

Simplify

\frac{18 0 ^{\circ} + 18 β}{18} \cdot 18 : 18 0^{\circ} + 18 β

\frac{18 0 ^{\circ} + 18 β}{18} \cdot 18

Multiply fractions:

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

= \frac{( 18 0 ^{\circ} + 18 β ) \cdot 18}{18}

Cancel the common factor:

18

= 18 0^{\circ} + 18 β

Simplify

27 0^{\circ} \cdot 18 : 486 0^{\circ}

27 0^{\circ} \cdot 18

Multiply fractions:

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

= 486 0^{\circ}

Multiply the numbers:

3 \cdot 18 = 54

= 486 0^{\circ}

Divide the numbers:

\frac{54}{2} = 27

= 486 0^{\circ}

Simplify

36 0^{\circ} n \cdot 18 : 648 0^{\circ} n

36 0^{\circ} n \cdot 18

Multiply the numbers:

2 \cdot 18 = 36

= 648 0^{\circ} n

- 18 0^{\circ} + 36 β + 18 0^{\circ} + 18 β = 486 0^{\circ} + 648 0^{\circ} n

54 β = 486 0^{\circ} + 648 0^{\circ} n

54 β = 486 0^{\circ} + 648 0^{\circ} n

54 β = 486 0^{\circ} + 648 0^{\circ} n

Divide both sides by

54

54 β = 486 0^{\circ} + 648 0^{\circ} n

Divide both sides by

54

\frac{54 β}{54} = 9 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Simplify

\frac{54 β}{54} = 9 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Simplify

\frac{54 β}{54} : β

\frac{54 β}{54}

Divide the numbers:

\frac{54}{54} = 1

= β

Simplify

9 0^{\circ} + \frac{648 0 ^{\circ} n}{54} : 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

9 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Cancel

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

9 0^{\circ}

Cancel the common factor:

27

= 9 0^{\circ}

= 9 0^{\circ} + \frac{648 0 ^{\circ} n}{54}

Cancel

\frac{648 0 ^{\circ} n}{54} : \frac{36 0 ^{\circ} n}{3}

\frac{648 0 ^{\circ} n}{54}

Cancel the common factor:

18

= \frac{36 0 ^{\circ} n}{3}

= 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

β = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{3}, β = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for tan(β+10)=cot(2β-10) ?
The general solution for tan(β+10)=cot(2β-10) is β=30+(360n)/3 ,β=90+(360n)/3