What is the general solution for tan(5x)*cot(x+40)=1 ?

The general solution for tan(5x)*cot(x+40)=1 is x=(180n)/2+10,x=55+(180n)/2

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tan(5x)*cot(x+40)=1

Solution

tan (5 x) \cdot cot (x + 4 0^{\circ}) = 1

Solution

x = \frac{18 0 ^{\circ} n}{2} + 1 0^{\circ}, x = 5 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

Radians

x = \frac{π}{18} + \frac{π}{2} n, x = \frac{11 π}{36} + \frac{π}{2} n

Solution steps

tan (5 x) cot (x + 4 0^{\circ}) = 1

Subtract

1

from both sides

tan (5 x) cot (x + 4 0^{\circ}) - 1 = 0

Express with sin, cos

- 1 + cot (4 0^{\circ} + x) tan (5 x)

Use the basic trigonometric identity:

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

= - 1 + \frac{cos ( 4 0 ^{\circ} + x )}{sin ( 4 0 ^{\circ} + x )} tan (5 x)

Use the basic trigonometric identity:

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

= - 1 + \frac{cos ( 4 0 ^{\circ} + x )}{sin ( 4 0 ^{\circ} + x )} \cdot \frac{sin ( 5 x )}{cos ( 5 x )}

Simplify

- 1 + \frac{cos ( 4 0 ^{\circ} + x )}{sin ( 4 0 ^{\circ} + x )} \cdot \frac{sin ( 5 x )}{cos ( 5 x )} : \frac{- sin ( \frac{36 0 ^{\circ} + 9 x}{9} ) cos ( 5 x ) + cos ( \frac{36 0 ^{\circ} + 9 x}{9} ) sin ( 5 x )}{sin ( \frac{36 0 ^{\circ} + 9 x}{9} ) cos ( 5 x )}

- 1 + \frac{cos ( 4 0 ^{\circ} + x )}{sin ( 4 0 ^{\circ} + x )} \cdot \frac{sin ( 5 x )}{cos ( 5 x )}

\frac{cos ( 4 0 ^{\circ} + x )}{sin ( 4 0 ^{\circ} + x )} \cdot \frac{sin ( 5 x )}{cos ( 5 x )} = \frac{cos ( \frac{36 0 ^{\circ} + 9 x}{9} ) sin ( 5 x )}{sin ( \frac{36 0 ^{\circ} + 9 x}{9} ) cos ( 5 x )}

\frac{cos ( 4 0 ^{\circ} + x )}{sin ( 4 0 ^{\circ} + x )} \cdot \frac{sin ( 5 x )}{cos ( 5 x )}

Multiply fractions:

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

= \frac{cos ( 4 0 ^{\circ} + x ) sin ( 5 x )}{sin ( 4 0 ^{\circ} + x ) cos ( 5 x )}

Join

4 0^{\circ} + x : \frac{36 0 ^{\circ} + 9 x}{9}

4 0^{\circ} + x

Convert element to fraction:

x = \frac{x 9}{9}

= 4 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{36 0 ^{\circ} + x \cdot 9}{9}

= \frac{cos ( x + 4 0 ^{\circ} ) sin ( 5 x )}{sin ( \frac{9 x + 36 0 ^{\circ}}{9} ) cos ( 5 x )}

Join

4 0^{\circ} + x : \frac{36 0 ^{\circ} + 9 x}{9}

4 0^{\circ} + x

Convert element to fraction:

x = \frac{x 9}{9}

= 4 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{36 0 ^{\circ} + x \cdot 9}{9}

= \frac{cos ( \frac{9 x + 36 0 ^{\circ}}{9} ) sin ( 5 x )}{sin ( \frac{9 x + 36 0 ^{\circ}}{9} ) cos ( 5 x )}

= - 1 + \frac{cos ( \frac{9 x + 36 0 ^{\circ}}{9} ) sin ( 5 x )}{sin ( \frac{9 x + 36 0 ^{\circ}}{9} ) cos ( 5 x )}

Convert element to fraction:

1 = \frac{1 sin ( \frac{36 0 ^{\circ} + x 9}{9} ) cos ( 5 x )}{sin ( \frac{36 0 ^{\circ} + x 9}{9} ) cos ( 5 x )}

= - \frac{1 \cdot sin ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) cos ( 5 x )}{sin ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) cos ( 5 x )} + \frac{cos ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) sin ( 5 x )}{sin ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) cos ( 5 x )}

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{36 0 ^{\circ} + x \cdot 9}{9} ) cos ( 5 x ) + cos ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) sin ( 5 x )}{sin ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) cos ( 5 x )}

Multiply:

1 \cdot sin (\frac{36 0 ^{\circ} + x \cdot 9}{9}) = sin (\frac{36 0 ^{\circ} + x \cdot 9}{9})

= \frac{- sin ( \frac{9 x + 36 0 ^{\circ}}{9} ) cos ( 5 x ) + cos ( \frac{9 x + 36 0 ^{\circ}}{9} ) sin ( 5 x )}{sin ( \frac{9 x + 36 0 ^{\circ}}{9} ) cos ( 5 x )}

= \frac{- sin ( \frac{36 0 ^{\circ} + 9 x}{9} ) cos ( 5 x ) + cos ( \frac{36 0 ^{\circ} + 9 x}{9} ) sin ( 5 x )}{sin ( \frac{36 0 ^{\circ} + 9 x}{9} ) cos ( 5 x )}

\frac{- cos ( 5 x ) sin ( \frac{36 0 ^{\circ} + 9 x}{9} ) + cos ( \frac{36 0 ^{\circ} + 9 x}{9} ) sin ( 5 x )}{cos ( 5 x ) sin ( \frac{36 0 ^{\circ} + 9 x}{9} )} = 0

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

- cos (5 x) sin (\frac{36 0 ^{\circ} + 9 x}{9}) + cos (\frac{36 0 ^{\circ} + 9 x}{9}) sin (5 x) = 0

Rewrite using trig identities

- cos (5 x) sin (\frac{36 0 ^{\circ} + 9 x}{9}) + cos (\frac{36 0 ^{\circ} + 9 x}{9}) sin (5 x)

Use the Angle Difference identity:

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

= sin (5 x - \frac{36 0 ^{\circ} + 9 x}{9})

sin (5 x - \frac{36 0 ^{\circ} + 9 x}{9}) = 0

General solutions for

sin (5 x - \frac{36 0 ^{\circ} + 9 x}{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}

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

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

Solve

5 x - \frac{36 0 ^{\circ} + 9 x}{9} = 0 + 36 0^{\circ} n : x = \frac{18 0 ^{\circ} n}{2} + 1 0^{\circ}

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

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

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

Multiply both sides by

9

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

Multiply both sides by

9

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

Simplify

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

Simplify

5 x \cdot 9 : 45 x

5 x \cdot 9

Multiply the numbers:

5 \cdot 9 = 45

= 45 x

Simplify

- \frac{36 0 ^{\circ} + 9 x}{9} \cdot 9 : - (36 0^{\circ} + 9 x)

- \frac{36 0 ^{\circ} + 9 x}{9} \cdot 9

Multiply fractions:

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

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

Cancel the common factor:

9

= - (9 x + 36 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

45 x - (36 0^{\circ} + 9 x) = 324 0^{\circ} n

45 x - (36 0^{\circ} + 9 x) = 324 0^{\circ} n

45 x - (36 0^{\circ} + 9 x) = 324 0^{\circ} n

Expand

45 x - (36 0^{\circ} + 9 x) : 36 x - 36 0^{\circ}

45 x - (36 0^{\circ} + 9 x)

- (36 0^{\circ} + 9 x) : - 36 0^{\circ} - 9 x

- (36 0^{\circ} + 9 x)

Distribute parentheses

= - (36 0^{\circ}) - (9 x)

Apply minus-plus rules

+ (- a) = - a

= - 36 0^{\circ} - 9 x

= 45 x - 36 0^{\circ} - 9 x

Simplify

45 x - 36 0^{\circ} - 9 x : 36 x - 36 0^{\circ}

45 x - 36 0^{\circ} - 9 x

Group like terms

= 45 x - 9 x - 36 0^{\circ}

Add similar elements:

45 x - 9 x = 36 x

= 36 x - 36 0^{\circ}

= 36 x - 36 0^{\circ}

36 x - 36 0^{\circ} = 324 0^{\circ} n

Move

36 0^{\circ}

to the right side

36 x - 36 0^{\circ} = 324 0^{\circ} n

Add

36 0^{\circ}

to both sides

36 x - 36 0^{\circ} + 36 0^{\circ} = 324 0^{\circ} n + 36 0^{\circ}

Simplify

36 x = 324 0^{\circ} n + 36 0^{\circ}

36 x = 324 0^{\circ} n + 36 0^{\circ}

Divide both sides by

36

36 x = 324 0^{\circ} n + 36 0^{\circ}

Divide both sides by

36

\frac{36 x}{36} = \frac{324 0 ^{\circ} n}{36} + 1 0^{\circ}

Simplify

\frac{36 x}{36} = \frac{324 0 ^{\circ} n}{36} + 1 0^{\circ}

Simplify

\frac{36 x}{36} : x

\frac{36 x}{36}

Divide the numbers:

\frac{36}{36} = 1

= x

Simplify

\frac{324 0 ^{\circ} n}{36} + 1 0^{\circ} : \frac{18 0 ^{\circ} n}{2} + 1 0^{\circ}

\frac{324 0 ^{\circ} n}{36} + 1 0^{\circ}

Cancel

\frac{324 0 ^{\circ} n}{36} : \frac{18 0 ^{\circ} n}{2}

\frac{324 0 ^{\circ} n}{36}

Cancel the common factor:

18

= \frac{18 0 ^{\circ} n}{2}

= \frac{18 0 ^{\circ} n}{2} + 1 0^{\circ}

Cancel

1 0^{\circ} : 1 0^{\circ}

1 0^{\circ}

Cancel the common factor:

2

= 1 0^{\circ}

= \frac{18 0 ^{\circ} n}{2} + 1 0^{\circ}

x = \frac{18 0 ^{\circ} n}{2} + 1 0^{\circ}

x = \frac{18 0 ^{\circ} n}{2} + 1 0^{\circ}

x = \frac{18 0 ^{\circ} n}{2} + 1 0^{\circ}

Solve

5 x - \frac{36 0 ^{\circ} + 9 x}{9} = 18 0^{\circ} + 36 0^{\circ} n : x = 5 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

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

Multiply both sides by

9

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

Multiply both sides by

9

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

Simplify

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

Simplify

5 x \cdot 9 : 45 x

5 x \cdot 9

Multiply the numbers:

5 \cdot 9 = 45

= 45 x

Simplify

- \frac{36 0 ^{\circ} + 9 x}{9} \cdot 9 : - (36 0^{\circ} + 9 x)

- \frac{36 0 ^{\circ} + 9 x}{9} \cdot 9

Multiply fractions:

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

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

Cancel the common factor:

9

= - (9 x + 36 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

45 x - (36 0^{\circ} + 9 x) = 162 0^{\circ} + 324 0^{\circ} n

45 x - (36 0^{\circ} + 9 x) = 162 0^{\circ} + 324 0^{\circ} n

45 x - (36 0^{\circ} + 9 x) = 162 0^{\circ} + 324 0^{\circ} n

Expand

45 x - (36 0^{\circ} + 9 x) : 36 x - 36 0^{\circ}

45 x - (36 0^{\circ} + 9 x)

- (36 0^{\circ} + 9 x) : - 36 0^{\circ} - 9 x

- (36 0^{\circ} + 9 x)

Distribute parentheses

= - (36 0^{\circ}) - (9 x)

Apply minus-plus rules

+ (- a) = - a

= - 36 0^{\circ} - 9 x

= 45 x - 36 0^{\circ} - 9 x

Simplify

45 x - 36 0^{\circ} - 9 x : 36 x - 36 0^{\circ}

45 x - 36 0^{\circ} - 9 x

Group like terms

= 45 x - 9 x - 36 0^{\circ}

Add similar elements:

45 x - 9 x = 36 x

= 36 x - 36 0^{\circ}

= 36 x - 36 0^{\circ}

36 x - 36 0^{\circ} = 162 0^{\circ} + 324 0^{\circ} n

Move

36 0^{\circ}

to the right side

36 x - 36 0^{\circ} = 162 0^{\circ} + 324 0^{\circ} n

Add

36 0^{\circ}

to both sides

36 x - 36 0^{\circ} + 36 0^{\circ} = 162 0^{\circ} + 324 0^{\circ} n + 36 0^{\circ}

Simplify

36 x = 198 0^{\circ} + 324 0^{\circ} n

36 x = 198 0^{\circ} + 324 0^{\circ} n

Divide both sides by

36

36 x = 198 0^{\circ} + 324 0^{\circ} n

Divide both sides by

36

\frac{36 x}{36} = 5 5^{\circ} + \frac{324 0 ^{\circ} n}{36}

Simplify

x = 5 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

x = 5 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

x = \frac{18 0 ^{\circ} n}{2} + 1 0^{\circ}, x = 5 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for tan(5x)*cot(x+40)=1 ?
The general solution for tan(5x)*cot(x+40)=1 is x=(180n)/2+10,x=55+(180n)/2