What is the general solution for tan(2a)cot(a+20)=1 ?

The general solution for tan(2a)cot(a+20)=1 is a=360n+20,a=200+360n

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tan(2a)cot(a+20)=1

Solution

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

Solution

a = 36 0^{\circ} n + 2 0^{\circ}, a = 20 0^{\circ} + 36 0^{\circ} n

Radians

a = \frac{π}{9} + 2 πn, a = \frac{10 π}{9} + 2 πn

Solution steps

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

Subtract

1

from both sides

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

= - 1 + \frac{cos ( 2 0 ^{\circ} + a )}{sin ( 2 0 ^{\circ} + a )} tan (2 a)

Use the basic trigonometric identity:

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

= - 1 + \frac{cos ( 2 0 ^{\circ} + a )}{sin ( 2 0 ^{\circ} + a )} \cdot \frac{sin ( 2 a )}{cos ( 2 a )}

Simplify

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

- 1 + \frac{cos ( 2 0 ^{\circ} + a )}{sin ( 2 0 ^{\circ} + a )} \cdot \frac{sin ( 2 a )}{cos ( 2 a )}

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

\frac{cos ( 2 0 ^{\circ} + a )}{sin ( 2 0 ^{\circ} + a )} \cdot \frac{sin ( 2 a )}{cos ( 2 a )}

Multiply fractions:

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

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

Join

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

2 0^{\circ} + a

Convert element to fraction:

a = \frac{a 9}{9}

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

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

Join

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

2 0^{\circ} + a

Convert element to fraction:

a = \frac{a 9}{9}

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

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

= - 1 + \frac{cos ( \frac{9 a + 18 0 ^{\circ}}{9} ) sin ( 2 a )}{sin ( \frac{9 a + 18 0 ^{\circ}}{9} ) cos ( 2 a )}

Convert element to fraction:

1 = \frac{1 sin ( \frac{18 0 ^{\circ} + a 9}{9} ) cos ( 2 a )}{sin ( \frac{18 0 ^{\circ} + a 9}{9} ) cos ( 2 a )}

= - \frac{1 \cdot sin ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) cos ( 2 a )}{sin ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) cos ( 2 a )} + \frac{cos ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) sin ( 2 a )}{sin ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) cos ( 2 a )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot sin ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) cos ( 2 a ) + cos ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) sin ( 2 a )}{sin ( \frac{18 0 ^{\circ} + a \cdot 9}{9} ) cos ( 2 a )}

Multiply:

1 \cdot sin (\frac{18 0 ^{\circ} + a \cdot 9}{9}) = sin (\frac{18 0 ^{\circ} + a \cdot 9}{9})

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

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

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

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

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

Rewrite using trig identities

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

Use the Angle Difference identity:

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

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

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

General solutions for

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

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 0 + 36 0^{\circ} n, 2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 18 0^{\circ} + 36 0^{\circ} n

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 0 + 36 0^{\circ} n, 2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 18 0^{\circ} + 36 0^{\circ} n

Solve

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 0 + 36 0^{\circ} n : a = 36 0^{\circ} n + 2 0^{\circ}

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

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

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

Multiply both sides by

9

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

Multiply both sides by

9

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

Simplify

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

Simplify

2 a \cdot 9 : 18 a

2 a \cdot 9

Multiply the numbers:

2 \cdot 9 = 18

= 18 a

Simplify

- \frac{18 0 ^{\circ} + 9 a}{9} \cdot 9 : - (18 0^{\circ} + 9 a)

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

Multiply fractions:

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

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

Cancel the common factor:

9

= - (9 a + 18 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 a - (18 0^{\circ} + 9 a) = 324 0^{\circ} n

18 a - (18 0^{\circ} + 9 a) = 324 0^{\circ} n

18 a - (18 0^{\circ} + 9 a) = 324 0^{\circ} n

Expand

18 a - (18 0^{\circ} + 9 a) : 9 a - 18 0^{\circ}

18 a - (18 0^{\circ} + 9 a)

- (18 0^{\circ} + 9 a) : - 18 0^{\circ} - 9 a

- (18 0^{\circ} + 9 a)

Distribute parentheses

= - (18 0^{\circ}) - (9 a)

Apply minus-plus rules

+ (- a) = - a

= - 18 0^{\circ} - 9 a

= 18 a - 18 0^{\circ} - 9 a

Simplify

18 a - 18 0^{\circ} - 9 a : 9 a - 18 0^{\circ}

18 a - 18 0^{\circ} - 9 a

Group like terms

= 18 a - 9 a - 18 0^{\circ}

Add similar elements:

18 a - 9 a = 9 a

= 9 a - 18 0^{\circ}

= 9 a - 18 0^{\circ}

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

Move

18 0^{\circ}

to the right side

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

Add

18 0^{\circ}

to both sides

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

Simplify

9 a = 324 0^{\circ} n + 18 0^{\circ}

9 a = 324 0^{\circ} n + 18 0^{\circ}

Divide both sides by

9

9 a = 324 0^{\circ} n + 18 0^{\circ}

Divide both sides by

9

\frac{9 a}{9} = \frac{324 0 ^{\circ} n}{9} + 2 0^{\circ}

Simplify

a = 36 0^{\circ} n + 2 0^{\circ}

a = 36 0^{\circ} n + 2 0^{\circ}

Solve

2 a - \frac{18 0 ^{\circ} + 9 a}{9} = 18 0^{\circ} + 36 0^{\circ} n : a = 20 0^{\circ} + 36 0^{\circ} n

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

Multiply both sides by

9

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

Multiply both sides by

9

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

Simplify

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

Simplify

2 a \cdot 9 : 18 a

2 a \cdot 9

Multiply the numbers:

2 \cdot 9 = 18

= 18 a

Simplify

- \frac{18 0 ^{\circ} + 9 a}{9} \cdot 9 : - (18 0^{\circ} + 9 a)

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

Multiply fractions:

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

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

Cancel the common factor:

9

= - (9 a + 18 0^{\circ})

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

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

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

Expand

18 a - (18 0^{\circ} + 9 a) : 9 a - 18 0^{\circ}

18 a - (18 0^{\circ} + 9 a)

- (18 0^{\circ} + 9 a) : - 18 0^{\circ} - 9 a

- (18 0^{\circ} + 9 a)

Distribute parentheses

= - (18 0^{\circ}) - (9 a)

Apply minus-plus rules

+ (- a) = - a

= - 18 0^{\circ} - 9 a

= 18 a - 18 0^{\circ} - 9 a

Simplify

18 a - 18 0^{\circ} - 9 a : 9 a - 18 0^{\circ}

18 a - 18 0^{\circ} - 9 a

Group like terms

= 18 a - 9 a - 18 0^{\circ}

Add similar elements:

18 a - 9 a = 9 a

= 9 a - 18 0^{\circ}

= 9 a - 18 0^{\circ}

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

Move

18 0^{\circ}

to the right side

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

Add

18 0^{\circ}

to both sides

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

Simplify

9 a = 180 0^{\circ} + 324 0^{\circ} n

9 a = 180 0^{\circ} + 324 0^{\circ} n

Divide both sides by

9

9 a = 180 0^{\circ} + 324 0^{\circ} n

Divide both sides by

9

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

Simplify

a = 20 0^{\circ} + 36 0^{\circ} n

a = 20 0^{\circ} + 36 0^{\circ} n

a = 36 0^{\circ} n + 2 0^{\circ}, a = 20 0^{\circ} + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for tan(2a)cot(a+20)=1 ?
The general solution for tan(2a)cot(a+20)=1 is a=360n+20,a=200+360n