What is the general solution for tan(θ+32)=cot(θ-20) ?

The general solution for tan(θ+32)=cot(θ-20) is θ=39+180n,θ=129+180n

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Popular Trigonometry >

tan(θ+32)=cot(θ-20)

Solution

tan (θ + 3 2^{\circ}) = cot (θ - 2 0^{\circ})

Solution

θ = 3 9^{\circ} + 18 0^{\circ} n, θ = 12 9^{\circ} + 18 0^{\circ} n

Radians

θ = \frac{13 π}{60} + πn, θ = \frac{43 π}{60} + πn

Solution steps

tan (θ + 3 2^{\circ}) = cot (θ - 2 0^{\circ})

Subtract

cot (θ - 2 0^{\circ})

from both sides

tan (θ + 3 2^{\circ}) - cot (θ - 2 0^{\circ}) = 0

Simplify

tan (θ + 3 2^{\circ}) - cot (θ - 2 0^{\circ}) : tan (\frac{45 θ + 144 0 ^{\circ}}{45}) - cot (\frac{9 θ - 18 0 ^{\circ}}{9})

tan (θ + 3 2^{\circ}) - cot (θ - 2 0^{\circ})

Join

θ + 3 2^{\circ} : \frac{45 θ + 144 0 ^{\circ}}{45}

θ + 3 2^{\circ}

Convert element to fraction:

θ = \frac{θ 45}{45}

= \frac{θ \cdot 45}{45} + 3 2^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{θ \cdot 45 + 144 0 ^{\circ}}{45}

= tan (\frac{45 θ + 144 0 ^{\circ}}{45}) - cot (θ - 2 0^{\circ})

Join

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

θ - 2 0^{\circ}

Convert element to fraction:

θ = \frac{θ 9}{9}

= \frac{θ \cdot 9}{9} - 2 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{θ \cdot 9 - 18 0 ^{\circ}}{9}

= tan (\frac{45 θ + 144 0 ^{\circ}}{45}) - cot (\frac{9 θ - 18 0 ^{\circ}}{9})

tan (\frac{45 θ + 144 0 ^{\circ}}{45}) - cot (\frac{9 θ - 18 0 ^{\circ}}{9}) = 0

Express with sin, cos

- cot (\frac{- 18 0 ^{\circ} + 9 θ}{9}) + tan (\frac{45 θ + 144 0 ^{\circ}}{45})

Use the basic trigonometric identity:

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

= - \frac{cos ( \frac{- 18 0 ^{\circ} + 9 θ}{9} )}{sin ( \frac{- 18 0 ^{\circ} + 9 θ}{9} )} + tan (\frac{45 θ + 144 0 ^{\circ}}{45})

Use the basic trigonometric identity:

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

= - \frac{cos ( \frac{- 18 0 ^{\circ} + 9 θ}{9} )}{sin ( \frac{- 18 0 ^{\circ} + 9 θ}{9} )} + \frac{sin ( \frac{45 θ + 144 0 ^{\circ}}{45} )}{cos ( \frac{45 θ + 144 0 ^{\circ}}{45} )}

Simplify

- \frac{cos ( \frac{- 18 0 ^{\circ} + 9 θ}{9} )}{sin ( \frac{- 18 0 ^{\circ} + 9 θ}{9} )} + \frac{sin ( \frac{45 θ + 144 0 ^{\circ}}{45} )}{cos ( \frac{45 θ + 144 0 ^{\circ}}{45} )} : \frac{- cos ( \frac{- 18 0 ^{\circ} + 9 θ}{9} ) cos ( \frac{45 θ + 144 0 ^{\circ}}{45} ) + sin ( \frac{45 θ + 144 0 ^{\circ}}{45} ) sin ( \frac{9 θ - 18 0 ^{\circ}}{9} )}{sin ( \frac{9 θ - 18 0 ^{\circ}}{9} ) cos ( \frac{45 θ + 144 0 ^{\circ}}{45} )}

- \frac{cos ( \frac{- 18 0 ^{\circ} + 9 θ}{9} )}{sin ( \frac{- 18 0 ^{\circ} + 9 θ}{9} )} + \frac{sin ( \frac{45 θ + 144 0 ^{\circ}}{45} )}{cos ( \frac{45 θ + 144 0 ^{\circ}}{45} )}

Least Common Multiplier of

sin (\frac{- 18 0 ^{\circ} + 9 θ}{9}), cos (\frac{45 θ + 144 0 ^{\circ}}{45}) : sin (\frac{9 θ - 18 0 ^{\circ}}{9}) cos (\frac{45 θ + 144 0 ^{\circ}}{45})

sin (\frac{- 18 0 ^{\circ} + 9 θ}{9}), cos (\frac{45 θ + 144 0 ^{\circ}}{45})

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (\frac{- 18 0 ^{\circ} + 9 θ}{9})

cos (\frac{45 θ + 144 0 ^{\circ}}{45})

= sin (\frac{9 θ - 18 0 ^{\circ}}{9}) cos (\frac{45 θ + 144 0 ^{\circ}}{45})

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{9 θ - 18 0 ^{\circ}}{9}) cos (\frac{45 θ + 144 0 ^{\circ}}{45})

For

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

multiply the denominator and numerator by

cos (\frac{45 θ + 144 0 ^{\circ}}{45})

\frac{cos ( \frac{- 18 0 ^{\circ} + 9 θ}{9} )}{sin ( \frac{- 18 0 ^{\circ} + 9 θ}{9} )} = \frac{cos ( \frac{- 18 0 ^{\circ} + 9 θ}{9} ) cos ( \frac{45 θ + 144 0 ^{\circ}}{45} )}{sin ( \frac{- 18 0 ^{\circ} + 9 θ}{9} ) cos ( \frac{45 θ + 144 0 ^{\circ}}{45} )}

For

\frac{sin ( \frac{45 θ + 144 0 ^{\circ}}{45} )}{cos ( \frac{45 θ + 144 0 ^{\circ}}{45} )} :

multiply the denominator and numerator by

sin (\frac{9 θ - 18 0 ^{\circ}}{9})

\frac{sin ( \frac{45 θ + 144 0 ^{\circ}}{45} )}{cos ( \frac{45 θ + 144 0 ^{\circ}}{45} )} = \frac{sin ( \frac{45 θ + 144 0 ^{\circ}}{45} ) sin ( \frac{9 θ - 18 0 ^{\circ}}{9} )}{cos ( \frac{45 θ + 144 0 ^{\circ}}{45} ) sin ( \frac{9 θ - 18 0 ^{\circ}}{9} )}

= - \frac{cos ( \frac{- 18 0 ^{\circ} + 9 θ}{9} ) cos ( \frac{45 θ + 144 0 ^{\circ}}{45} )}{sin ( \frac{- 18 0 ^{\circ} + 9 θ}{9} ) cos ( \frac{45 θ + 144 0 ^{\circ}}{45} )} + \frac{sin ( \frac{45 θ + 144 0 ^{\circ}}{45} ) sin ( \frac{9 θ - 18 0 ^{\circ}}{9} )}{cos ( \frac{45 θ + 144 0 ^{\circ}}{45} ) sin ( \frac{9 θ - 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{- cos ( \frac{- 18 0 ^{\circ} + 9 θ}{9} ) cos ( \frac{45 θ + 144 0 ^{\circ}}{45} ) + sin ( \frac{45 θ + 144 0 ^{\circ}}{45} ) sin ( \frac{9 θ - 18 0 ^{\circ}}{9} )}{sin ( \frac{9 θ - 18 0 ^{\circ}}{9} ) cos ( \frac{45 θ + 144 0 ^{\circ}}{45} )}

= \frac{- cos ( \frac{- 18 0 ^{\circ} + 9 θ}{9} ) cos ( \frac{45 θ + 144 0 ^{\circ}}{45} ) + sin ( \frac{45 θ + 144 0 ^{\circ}}{45} ) sin ( \frac{9 θ - 18 0 ^{\circ}}{9} )}{sin ( \frac{9 θ - 18 0 ^{\circ}}{9} ) cos ( \frac{45 θ + 144 0 ^{\circ}}{45} )}

\frac{- cos ( \frac{- 18 0 ^{\circ} + 9 θ}{9} ) cos ( \frac{45 θ + 144 0 ^{\circ}}{45} ) + sin ( \frac{- 18 0 ^{\circ} + 9 θ}{9} ) sin ( \frac{45 θ + 144 0 ^{\circ}}{45} )}{cos ( \frac{45 θ + 144 0 ^{\circ}}{45} ) sin ( \frac{- 18 0 ^{\circ} + 9 θ}{9} )} = 0

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

- cos (\frac{- 18 0 ^{\circ} + 9 θ}{9}) cos (\frac{45 θ + 144 0 ^{\circ}}{45}) + sin (\frac{- 18 0 ^{\circ} + 9 θ}{9}) sin (\frac{45 θ + 144 0 ^{\circ}}{45}) = 0

Rewrite using trig identities

- cos (\frac{- 18 0 ^{\circ} + 9 θ}{9}) cos (\frac{45 θ + 144 0 ^{\circ}}{45}) + sin (\frac{- 18 0 ^{\circ} + 9 θ}{9}) sin (\frac{45 θ + 144 0 ^{\circ}}{45})

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 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45})

- cos (\frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45}) = 0

Divide both sides by

- 1

- cos (\frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45}) = 0

Divide both sides by

- 1

\frac{- cos ( \frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45} )}{- 1} = \frac{0}{- 1}

Simplify

cos (\frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45}) = 0

cos (\frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45}) = 0

General solutions for

cos (\frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45}) = 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 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45} = 9 0^{\circ} + 36 0^{\circ} n, \frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45} = 27 0^{\circ} + 36 0^{\circ} n

\frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45} = 9 0^{\circ} + 36 0^{\circ} n, \frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45} = 27 0^{\circ} + 36 0^{\circ} n

Solve

\frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45} = 9 0^{\circ} + 36 0^{\circ} n : θ = 3 9^{\circ} + 18 0^{\circ} n

\frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45} = 9 0^{\circ} + 36 0^{\circ} n

Multiply by LCM

\frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45} = 9 0^{\circ} + 36 0^{\circ} n

Find Least Common Multiplier of

9, 45, 2 : 90

9, 45, 2

Least Common Multiplier (LCM)

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Prime factorization of

45 : 3 \cdot 3 \cdot 5

45

45

divides by

345 = 15 \cdot 3

= 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 3 \cdot 5

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Compute a number comprised of factors that appear in at least one of the following:

9, 45, 2

= 3 \cdot 3 \cdot 5 \cdot 2

Multiply the numbers:

3 \cdot 3 \cdot 5 \cdot 2 = 90

= 90

Multiply by LCM=

90

\frac{- 18 0 ^{\circ} + 9 θ}{9} \cdot 90 + \frac{45 θ + 144 0 ^{\circ}}{45} \cdot 90 = 9 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

Simplify

\frac{- 18 0 ^{\circ} + 9 θ}{9} \cdot 90 + \frac{45 θ + 144 0 ^{\circ}}{45} \cdot 90 = 9 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

Simplify

\frac{- 18 0 ^{\circ} + 9 θ}{9} \cdot 90 : 10 (9 θ - 18 0^{\circ})

\frac{- 18 0 ^{\circ} + 9 θ}{9} \cdot 90

Multiply fractions:

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

= \frac{( - 18 0 ^{\circ} + 9 θ ) \cdot 90}{9}

Divide the numbers:

\frac{90}{9} = 10

= 10 (9 θ - 18 0^{\circ})

Simplify

\frac{45 θ + 144 0 ^{\circ}}{45} \cdot 90 : 2 (45 θ + 144 0^{\circ})

\frac{45 θ + 144 0 ^{\circ}}{45} \cdot 90

Multiply fractions:

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

= \frac{( 45 θ + 144 0 ^{\circ} ) \cdot 90}{45}

Divide the numbers:

\frac{90}{45} = 2

= 2 (45 θ + 144 0^{\circ})

Simplify

9 0^{\circ} \cdot 90 : 810 0^{\circ}

9 0^{\circ} \cdot 90

Multiply fractions:

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

= 810 0^{\circ}

Divide the numbers:

\frac{90}{2} = 45

= 810 0^{\circ}

Simplify

36 0^{\circ} n \cdot 90 : 3240 0^{\circ} n

36 0^{\circ} n \cdot 90

Multiply the numbers:

2 \cdot 90 = 180

= 3240 0^{\circ} n

10 (9 θ - 18 0^{\circ}) + 2 (45 θ + 144 0^{\circ}) = 810 0^{\circ} + 3240 0^{\circ} n

10 (9 θ - 18 0^{\circ}) + 2 (45 θ + 144 0^{\circ}) = 810 0^{\circ} + 3240 0^{\circ} n

10 (9 θ - 18 0^{\circ}) + 2 (45 θ + 144 0^{\circ}) = 810 0^{\circ} + 3240 0^{\circ} n

Expand

10 (9 θ - 18 0^{\circ}) + 2 (45 θ + 144 0^{\circ}) : 180 θ + 108 0^{\circ}

10 (9 θ - 18 0^{\circ}) + 2 (45 θ + 144 0^{\circ})

Expand

10 (9 θ - 18 0^{\circ}) : 90 θ - 180 0^{\circ}

10 (9 θ - 18 0^{\circ})

Apply the distributive law:

a (b - c) = ab - a c

a = 10, b = 9 θ, c = 18 0^{\circ}

= 10 \cdot 9 θ - 180 0^{\circ}

Multiply the numbers:

10 \cdot 9 = 90

= 90 θ - 180 0^{\circ}

= 90 θ - 180 0^{\circ} + 2 (45 θ + 144 0^{\circ})

Expand

2 (45 θ + 144 0^{\circ}) : 90 θ + 288 0^{\circ}

2 (45 θ + 144 0^{\circ})

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 45 θ, c = 144 0^{\circ}

= 2 \cdot 45 θ + 2 \cdot 144 0^{\circ}

Simplify

2 \cdot 45 θ + 2 \cdot 144 0^{\circ} : 90 θ + 288 0^{\circ}

2 \cdot 45 θ + 2 \cdot 144 0^{\circ}

Multiply the numbers:

2 \cdot 45 = 90

= 90 θ + 2 \cdot 144 0^{\circ}

Multiply the numbers:

2 \cdot 8 = 16

= 90 θ + 288 0^{\circ}

= 90 θ + 288 0^{\circ}

= 90 θ - 180 0^{\circ} + 90 θ + 288 0^{\circ}

Simplify

90 θ - 180 0^{\circ} + 90 θ + 288 0^{\circ} : 180 θ + 108 0^{\circ}

90 θ - 180 0^{\circ} + 90 θ + 288 0^{\circ}

Group like terms

= 90 θ + 90 θ - 180 0^{\circ} + 288 0^{\circ}

Add similar elements:

90 θ + 90 θ = 180 θ

= 180 θ - 180 0^{\circ} + 288 0^{\circ}

Add similar elements:

- 180 0^{\circ} + 288 0^{\circ} = 108 0^{\circ}

= 180 θ + 108 0^{\circ}

= 180 θ + 108 0^{\circ}

180 θ + 108 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

Move

108 0^{\circ}

to the right side

180 θ + 108 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n

Subtract

108 0^{\circ}

from both sides

180 θ + 108 0^{\circ} - 108 0^{\circ} = 810 0^{\circ} + 3240 0^{\circ} n - 108 0^{\circ}

Simplify

180 θ = 702 0^{\circ} + 3240 0^{\circ} n

180 θ = 702 0^{\circ} + 3240 0^{\circ} n

Divide both sides by

180

180 θ = 702 0^{\circ} + 3240 0^{\circ} n

Divide both sides by

180

\frac{180 θ}{180} = 3 9^{\circ} + \frac{3240 0 ^{\circ} n}{180}

Simplify

\frac{180 θ}{180} = 3 9^{\circ} + \frac{3240 0 ^{\circ} n}{180}

Simplify

\frac{180 θ}{180} : θ

\frac{180 θ}{180}

Divide the numbers:

\frac{180}{180} = 1

= θ

Simplify

3 9^{\circ} + \frac{3240 0 ^{\circ} n}{180} : 3 9^{\circ} + 18 0^{\circ} n

3 9^{\circ} + \frac{3240 0 ^{\circ} n}{180}

Cancel

3 9^{\circ} : 3 9^{\circ}

3 9^{\circ}

Cancel the common factor:

3

= 3 9^{\circ}

= 3 9^{\circ} + \frac{3240 0 ^{\circ} n}{180}

Divide the numbers:

\frac{180}{180} = 1

= 3 9^{\circ} + 18 0^{\circ} n

θ = 3 9^{\circ} + 18 0^{\circ} n

θ = 3 9^{\circ} + 18 0^{\circ} n

θ = 3 9^{\circ} + 18 0^{\circ} n

Solve

\frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45} = 27 0^{\circ} + 36 0^{\circ} n : θ = 12 9^{\circ} + 18 0^{\circ} n

\frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45} = 27 0^{\circ} + 36 0^{\circ} n

Multiply by LCM

\frac{- 18 0 ^{\circ} + 9 θ}{9} + \frac{45 θ + 144 0 ^{\circ}}{45} = 27 0^{\circ} + 36 0^{\circ} n

Find Least Common Multiplier of

9, 45, 2 : 90

9, 45, 2

Least Common Multiplier (LCM)

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Prime factorization of

45 : 3 \cdot 3 \cdot 5

45

45

divides by

345 = 15 \cdot 3

= 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 3 \cdot 5

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Compute a number comprised of factors that appear in at least one of the following:

9, 45, 2

= 3 \cdot 3 \cdot 5 \cdot 2

Multiply the numbers:

3 \cdot 3 \cdot 5 \cdot 2 = 90

= 90

Multiply by LCM=

90

\frac{- 18 0 ^{\circ} + 9 θ}{9} \cdot 90 + \frac{45 θ + 144 0 ^{\circ}}{45} \cdot 90 = 27 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

Simplify

\frac{- 18 0 ^{\circ} + 9 θ}{9} \cdot 90 + \frac{45 θ + 144 0 ^{\circ}}{45} \cdot 90 = 27 0^{\circ} \cdot 90 + 36 0^{\circ} n \cdot 90

Simplify

\frac{- 18 0 ^{\circ} + 9 θ}{9} \cdot 90 : 10 (9 θ - 18 0^{\circ})

\frac{- 18 0 ^{\circ} + 9 θ}{9} \cdot 90

Multiply fractions:

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

= \frac{( - 18 0 ^{\circ} + 9 θ ) \cdot 90}{9}

Divide the numbers:

\frac{90}{9} = 10

= 10 (9 θ - 18 0^{\circ})

Simplify

\frac{45 θ + 144 0 ^{\circ}}{45} \cdot 90 : 2 (45 θ + 144 0^{\circ})

\frac{45 θ + 144 0 ^{\circ}}{45} \cdot 90

Multiply fractions:

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

= \frac{( 45 θ + 144 0 ^{\circ} ) \cdot 90}{45}

Divide the numbers:

\frac{90}{45} = 2

= 2 (45 θ + 144 0^{\circ})

Simplify

27 0^{\circ} \cdot 90 : 2430 0^{\circ}

27 0^{\circ} \cdot 90

Multiply fractions:

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

= 2430 0^{\circ}

Multiply the numbers:

3 \cdot 90 = 270

= 2430 0^{\circ}

Divide the numbers:

\frac{270}{2} = 135

= 2430 0^{\circ}

Simplify

36 0^{\circ} n \cdot 90 : 3240 0^{\circ} n

36 0^{\circ} n \cdot 90

Multiply the numbers:

2 \cdot 90 = 180

= 3240 0^{\circ} n

10 (9 θ - 18 0^{\circ}) + 2 (45 θ + 144 0^{\circ}) = 2430 0^{\circ} + 3240 0^{\circ} n

10 (9 θ - 18 0^{\circ}) + 2 (45 θ + 144 0^{\circ}) = 2430 0^{\circ} + 3240 0^{\circ} n

10 (9 θ - 18 0^{\circ}) + 2 (45 θ + 144 0^{\circ}) = 2430 0^{\circ} + 3240 0^{\circ} n

Expand

10 (9 θ - 18 0^{\circ}) + 2 (45 θ + 144 0^{\circ}) : 180 θ + 108 0^{\circ}

10 (9 θ - 18 0^{\circ}) + 2 (45 θ + 144 0^{\circ})

Expand

10 (9 θ - 18 0^{\circ}) : 90 θ - 180 0^{\circ}

10 (9 θ - 18 0^{\circ})

Apply the distributive law:

a (b - c) = ab - a c

a = 10, b = 9 θ, c = 18 0^{\circ}

= 10 \cdot 9 θ - 180 0^{\circ}

Multiply the numbers:

10 \cdot 9 = 90

= 90 θ - 180 0^{\circ}

= 90 θ - 180 0^{\circ} + 2 (45 θ + 144 0^{\circ})

Expand

2 (45 θ + 144 0^{\circ}) : 90 θ + 288 0^{\circ}

2 (45 θ + 144 0^{\circ})

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 45 θ, c = 144 0^{\circ}

= 2 \cdot 45 θ + 2 \cdot 144 0^{\circ}

Simplify

2 \cdot 45 θ + 2 \cdot 144 0^{\circ} : 90 θ + 288 0^{\circ}

2 \cdot 45 θ + 2 \cdot 144 0^{\circ}

Multiply the numbers:

2 \cdot 45 = 90

= 90 θ + 2 \cdot 144 0^{\circ}

Multiply the numbers:

2 \cdot 8 = 16

= 90 θ + 288 0^{\circ}

= 90 θ + 288 0^{\circ}

= 90 θ - 180 0^{\circ} + 90 θ + 288 0^{\circ}

Simplify

90 θ - 180 0^{\circ} + 90 θ + 288 0^{\circ} : 180 θ + 108 0^{\circ}

90 θ - 180 0^{\circ} + 90 θ + 288 0^{\circ}

Group like terms

= 90 θ + 90 θ - 180 0^{\circ} + 288 0^{\circ}

Add similar elements:

90 θ + 90 θ = 180 θ

= 180 θ - 180 0^{\circ} + 288 0^{\circ}

Add similar elements:

- 180 0^{\circ} + 288 0^{\circ} = 108 0^{\circ}

= 180 θ + 108 0^{\circ}

= 180 θ + 108 0^{\circ}

180 θ + 108 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

Move

108 0^{\circ}

to the right side

180 θ + 108 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n

Subtract

108 0^{\circ}

from both sides

180 θ + 108 0^{\circ} - 108 0^{\circ} = 2430 0^{\circ} + 3240 0^{\circ} n - 108 0^{\circ}

Simplify

180 θ = 2322 0^{\circ} + 3240 0^{\circ} n

180 θ = 2322 0^{\circ} + 3240 0^{\circ} n

Divide both sides by

180

180 θ = 2322 0^{\circ} + 3240 0^{\circ} n

Divide both sides by

180

\frac{180 θ}{180} = 12 9^{\circ} + \frac{3240 0 ^{\circ} n}{180}

Simplify

\frac{180 θ}{180} = 12 9^{\circ} + \frac{3240 0 ^{\circ} n}{180}

Simplify

\frac{180 θ}{180} : θ

\frac{180 θ}{180}

Divide the numbers:

\frac{180}{180} = 1

= θ

Simplify

12 9^{\circ} + \frac{3240 0 ^{\circ} n}{180} : 12 9^{\circ} + 18 0^{\circ} n

12 9^{\circ} + \frac{3240 0 ^{\circ} n}{180}

Cancel

12 9^{\circ} : 12 9^{\circ}

12 9^{\circ}

Cancel the common factor:

3

= 12 9^{\circ}

= 12 9^{\circ} + \frac{3240 0 ^{\circ} n}{180}

Divide the numbers:

\frac{180}{180} = 1

= 12 9^{\circ} + 18 0^{\circ} n

θ = 12 9^{\circ} + 18 0^{\circ} n

θ = 12 9^{\circ} + 18 0^{\circ} n

θ = 12 9^{\circ} + 18 0^{\circ} n

θ = 3 9^{\circ} + 18 0^{\circ} n, θ = 12 9^{\circ} + 18 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for tan(θ+32)=cot(θ-20) ?
The general solution for tan(θ+32)=cot(θ-20) is θ=39+180n,θ=129+180n