What is the general solution for tan(x-20)=cot(x+45) ?

The general solution for tan(x-20)=cot(x+45) is x=pin+pi/4-25/2 ,x=pin-25/2+(3pi)/4

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tan(x-20)=cot(x+45)

Solution

tan (x - 20) = cot (x + 45)

Solution

x = πn + \frac{π}{4} - \frac{25}{2}, x = πn - \frac{25}{2} + \frac{3 π}{4}

Degrees

x = - 671.19724 \dots^{\circ} + 18 0^{\circ} n, x = - 581.19724 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan (x - 20) = cot (x + 45)

Subtract

cot (x + 45)

from both sides

tan (x - 20) - cot (x + 45) = 0

Express with sin, cos

- cot (45 + x) + tan (- 20 + x)

Use the basic trigonometric identity:

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

= - \frac{cos ( 45 + x )}{sin ( 45 + x )} + tan (- 20 + x)

Use the basic trigonometric identity:

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

= - \frac{cos ( 45 + x )}{sin ( 45 + x )} + \frac{sin ( - 20 + x )}{cos ( - 20 + x )}

Simplify

- \frac{cos ( 45 + x )}{sin ( 45 + x )} + \frac{sin ( - 20 + x )}{cos ( - 20 + x )} : \frac{- cos ( 45 + x ) cos ( x - 20 ) + sin ( - 20 + x ) sin ( x + 45 )}{sin ( x + 45 ) cos ( x - 20 )}

- \frac{cos ( 45 + x )}{sin ( 45 + x )} + \frac{sin ( - 20 + x )}{cos ( - 20 + x )}

Least Common Multiplier of

sin (45 + x), cos (- 20 + x) : sin (x + 45) cos (x - 20)

sin (45 + x), cos (- 20 + x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (45 + x)

cos (- 20 + x)

= sin (x + 45) cos (x - 20)

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 + 45) cos (x - 20)

For

\frac{cos ( 45 + x )}{sin ( 45 + x )} :

multiply the denominator and numerator by

cos (x - 20)

\frac{cos ( 45 + x )}{sin ( 45 + x )} = \frac{cos ( 45 + x ) cos ( x - 20 )}{sin ( 45 + x ) cos ( x - 20 )}

For

\frac{sin ( - 20 + x )}{cos ( - 20 + x )} :

multiply the denominator and numerator by

sin (x + 45)

\frac{sin ( - 20 + x )}{cos ( - 20 + x )} = \frac{sin ( - 20 + x ) sin ( x + 45 )}{cos ( - 20 + x ) sin ( x + 45 )}

= - \frac{cos ( 45 + x ) cos ( x - 20 )}{sin ( 45 + x ) cos ( x - 20 )} + \frac{sin ( - 20 + x ) sin ( x + 45 )}{cos ( - 20 + x ) sin ( x + 45 )}

Since the denominators are equal, combine the fractions:

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

= \frac{- cos ( 45 + x ) cos ( x - 20 ) + sin ( - 20 + x ) sin ( x + 45 )}{sin ( x + 45 ) cos ( x - 20 )}

= \frac{- cos ( 45 + x ) cos ( x - 20 ) + sin ( - 20 + x ) sin ( x + 45 )}{sin ( x + 45 ) cos ( x - 20 )}

\frac{- cos ( - 20 + x ) cos ( 45 + x ) + sin ( - 20 + x ) sin ( 45 + x )}{cos ( - 20 + x ) sin ( 45 + x )} = 0

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

- cos (- 20 + x) cos (45 + x) + sin (- 20 + x) sin (45 + x) = 0

Rewrite using trig identities

- cos (- 20 + x) cos (45 + x) + sin (- 20 + x) sin (45 + 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 (- 20 + x + 45 + x)

- cos (- 20 + x + 45 + x) = 0

Divide both sides by

- 1

- cos (- 20 + x + 45 + x) = 0

Divide both sides by

- 1

\frac{- cos ( - 20 + x + 45 + x )}{- 1} = \frac{0}{- 1}

Simplify

cos (- 20 + x + 45 + x) = 0

cos (- 20 + x + 45 + x) = 0

General solutions for

cos (- 20 + x + 45 + x) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

- 20 + x + 45 + x = \frac{π}{2} + 2 πn, - 20 + x + 45 + x = \frac{3 π}{2} + 2 πn

- 20 + x + 45 + x = \frac{π}{2} + 2 πn, - 20 + x + 45 + x = \frac{3 π}{2} + 2 πn

Solve

- 20 + x + 45 + x = \frac{π}{2} + 2 πn : x = πn + \frac{π}{4} - \frac{25}{2}

- 20 + x + 45 + x = \frac{π}{2} + 2 πn

Group like terms

x + x - 20 + 45 = \frac{π}{2} + 2 πn

Add similar elements:

x + x = 2 x

2 x - 20 + 45 = \frac{π}{2} + 2 πn

Add/Subtract the numbers:

- 20 + 45 = 25

2 x + 25 = \frac{π}{2} + 2 πn

Move

25

to the right side

2 x + 25 = \frac{π}{2} + 2 πn

Subtract

25

from both sides

2 x + 25 - 25 = \frac{π}{2} + 2 πn - 25

Simplify

2 x = \frac{π}{2} + 2 πn - 25

2 x = \frac{π}{2} + 2 πn - 25

Divide both sides by

2

2 x = \frac{π}{2} + 2 πn - 25

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2} - \frac{25}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2} - \frac{25}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} - \frac{25}{2} : πn + \frac{π}{4} - \frac{25}{2}

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} - \frac{25}{2}

Group like terms

= - \frac{25}{2} + \frac{2 πn}{2} + \frac{\frac{π}{2}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

= - \frac{25}{2} + πn + \frac{π}{4}

Group like terms

= πn + \frac{π}{4} - \frac{25}{2}

x = πn + \frac{π}{4} - \frac{25}{2}

x = πn + \frac{π}{4} - \frac{25}{2}

x = πn + \frac{π}{4} - \frac{25}{2}

Solve

- 20 + x + 45 + x = \frac{3 π}{2} + 2 πn : x = πn - \frac{25}{2} + \frac{3 π}{4}

- 20 + x + 45 + x = \frac{3 π}{2} + 2 πn

Group like terms

x + x - 20 + 45 = \frac{3 π}{2} + 2 πn

Add similar elements:

x + x = 2 x

2 x - 20 + 45 = \frac{3 π}{2} + 2 πn

Add/Subtract the numbers:

- 20 + 45 = 25

2 x + 25 = \frac{3 π}{2} + 2 πn

Move

25

to the right side

2 x + 25 = \frac{3 π}{2} + 2 πn

Subtract

25

from both sides

2 x + 25 - 25 = \frac{3 π}{2} + 2 πn - 25

Simplify

2 x = \frac{3 π}{2} + 2 πn - 25

2 x = \frac{3 π}{2} + 2 πn - 25

Divide both sides by

2

2 x = \frac{3 π}{2} + 2 πn - 25

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2} - \frac{25}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2} - \frac{25}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2} - \frac{25}{2} : πn - \frac{25}{2} + \frac{3 π}{4}

\frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2} - \frac{25}{2}

Group like terms

= - \frac{25}{2} + \frac{2 πn}{2} + \frac{\frac{3 π}{2}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{3 π}{2}}{2} = \frac{3 π}{4}

\frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4}

= - \frac{25}{2} + πn + \frac{3 π}{4}

Group like terms

= πn - \frac{25}{2} + \frac{3 π}{4}

x = πn - \frac{25}{2} + \frac{3 π}{4}

x = πn - \frac{25}{2} + \frac{3 π}{4}

x = πn - \frac{25}{2} + \frac{3 π}{4}

x = πn + \frac{π}{4} - \frac{25}{2}, x = πn - \frac{25}{2} + \frac{3 π}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x-20)=cot(x+45) ?
The general solution for tan(x-20)=cot(x+45) is x=pin+pi/4-25/2 ,x=pin-25/2+(3pi)/4