What is the general solution for tan(3x-10)=cot(2x-40) ?

The general solution for tan(3x-10)=cot(2x-40) is x=10+(2pin)/5+pi/(10),x=10+(2pin)/5+(3pi)/(10)

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Popular Trigonometry >

tan(3x-10)=cot(2x-40)

Solution

tan (3 x - 10) = cot (2 x - 40)

Solution

x = 10 + \frac{2 πn}{5} + \frac{π}{10}, x = 10 + \frac{2 πn}{5} + \frac{3 π}{10}

Degrees

x = 590.95779 \dots^{\circ} + 7 2^{\circ} n, x = 626.95779 \dots^{\circ} + 7 2^{\circ} n

Solution steps

tan (3 x - 10) = cot (2 x - 40)

Subtract

cot (2 x - 40)

from both sides

tan (3 x - 10) - cot (2 x - 40) = 0

Express with sin, cos

- cot (- 40 + 2 x) + tan (- 10 + 3 x)

Use the basic trigonometric identity:

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

= - \frac{cos ( - 40 + 2 x )}{sin ( - 40 + 2 x )} + tan (- 10 + 3 x)

Use the basic trigonometric identity:

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

= - \frac{cos ( - 40 + 2 x )}{sin ( - 40 + 2 x )} + \frac{sin ( - 10 + 3 x )}{cos ( - 10 + 3 x )}

Simplify

- \frac{cos ( - 40 + 2 x )}{sin ( - 40 + 2 x )} + \frac{sin ( - 10 + 3 x )}{cos ( - 10 + 3 x )} : \frac{- cos ( - 40 + 2 x ) cos ( 3 x - 10 ) + sin ( - 10 + 3 x ) sin ( 2 x - 40 )}{sin ( 2 x - 40 ) cos ( 3 x - 10 )}

- \frac{cos ( - 40 + 2 x )}{sin ( - 40 + 2 x )} + \frac{sin ( - 10 + 3 x )}{cos ( - 10 + 3 x )}

Least Common Multiplier of

sin (- 40 + 2 x), cos (- 10 + 3 x) : sin (2 x - 40) cos (3 x - 10)

sin (- 40 + 2 x), cos (- 10 + 3 x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (- 40 + 2 x)

cos (- 10 + 3 x)

= sin (2 x - 40) cos (3 x - 10)

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 (2 x - 40) cos (3 x - 10)

For

\frac{cos ( - 40 + 2 x )}{sin ( - 40 + 2 x )} :

multiply the denominator and numerator by

cos (3 x - 10)

\frac{cos ( - 40 + 2 x )}{sin ( - 40 + 2 x )} = \frac{cos ( - 40 + 2 x ) cos ( 3 x - 10 )}{sin ( - 40 + 2 x ) cos ( 3 x - 10 )}

For

\frac{sin ( - 10 + 3 x )}{cos ( - 10 + 3 x )} :

multiply the denominator and numerator by

sin (2 x - 40)

\frac{sin ( - 10 + 3 x )}{cos ( - 10 + 3 x )} = \frac{sin ( - 10 + 3 x ) sin ( 2 x - 40 )}{cos ( - 10 + 3 x ) sin ( 2 x - 40 )}

= - \frac{cos ( - 40 + 2 x ) cos ( 3 x - 10 )}{sin ( - 40 + 2 x ) cos ( 3 x - 10 )} + \frac{sin ( - 10 + 3 x ) sin ( 2 x - 40 )}{cos ( - 10 + 3 x ) sin ( 2 x - 40 )}

Since the denominators are equal, combine the fractions:

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

= \frac{- cos ( - 40 + 2 x ) cos ( 3 x - 10 ) + sin ( - 10 + 3 x ) sin ( 2 x - 40 )}{sin ( 2 x - 40 ) cos ( 3 x - 10 )}

= \frac{- cos ( - 40 + 2 x ) cos ( 3 x - 10 ) + sin ( - 10 + 3 x ) sin ( 2 x - 40 )}{sin ( 2 x - 40 ) cos ( 3 x - 10 )}

\frac{- cos ( - 10 + 3 x ) cos ( - 40 + 2 x ) + sin ( - 10 + 3 x ) sin ( - 40 + 2 x )}{cos ( - 10 + 3 x ) sin ( - 40 + 2 x )} = 0

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

- cos (- 10 + 3 x) cos (- 40 + 2 x) + sin (- 10 + 3 x) sin (- 40 + 2 x) = 0

Rewrite using trig identities

- cos (- 10 + 3 x) cos (- 40 + 2 x) + sin (- 10 + 3 x) sin (- 40 + 2 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 (- 10 + 3 x - 40 + 2 x)

- cos (- 10 + 3 x - 40 + 2 x) = 0

Divide both sides by

- 1

- cos (- 10 + 3 x - 40 + 2 x) = 0

Divide both sides by

- 1

\frac{- cos ( - 10 + 3 x - 40 + 2 x )}{- 1} = \frac{0}{- 1}

Simplify

cos (- 10 + 3 x - 40 + 2 x) = 0

cos (- 10 + 3 x - 40 + 2 x) = 0

General solutions for

cos (- 10 + 3 x - 40 + 2 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}

- 10 + 3 x - 40 + 2 x = \frac{π}{2} + 2 πn, - 10 + 3 x - 40 + 2 x = \frac{3 π}{2} + 2 πn

- 10 + 3 x - 40 + 2 x = \frac{π}{2} + 2 πn, - 10 + 3 x - 40 + 2 x = \frac{3 π}{2} + 2 πn

Solve

- 10 + 3 x - 40 + 2 x = \frac{π}{2} + 2 πn : x = 10 + \frac{2 πn}{5} + \frac{π}{10}

- 10 + 3 x - 40 + 2 x = \frac{π}{2} + 2 πn

Group like terms

3 x + 2 x - 10 - 40 = \frac{π}{2} + 2 πn

Add similar elements:

3 x + 2 x = 5 x

5 x - 10 - 40 = \frac{π}{2} + 2 πn

Subtract the numbers:

- 10 - 40 = - 50

5 x - 50 = \frac{π}{2} + 2 πn

Move

50

to the right side

5 x - 50 = \frac{π}{2} + 2 πn

Add

50

to both sides

5 x - 50 + 50 = \frac{π}{2} + 2 πn + 50

Simplify

5 x = \frac{π}{2} + 2 πn + 50

5 x = \frac{π}{2} + 2 πn + 50

Divide both sides by

5

5 x = \frac{π}{2} + 2 πn + 50

Divide both sides by

5

\frac{5 x}{5} = \frac{\frac{π}{2}}{5} + \frac{2 πn}{5} + \frac{50}{5}

Simplify

\frac{5 x}{5} = \frac{\frac{π}{2}}{5} + \frac{2 πn}{5} + \frac{50}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{π}{2}}{5} + \frac{2 πn}{5} + \frac{50}{5} : 10 + \frac{2 πn}{5} + \frac{π}{10}

\frac{\frac{π}{2}}{5} + \frac{2 πn}{5} + \frac{50}{5}

Group like terms

= \frac{50}{5} + \frac{2 πn}{5} + \frac{\frac{π}{2}}{5}

\frac{50}{5} = 10

\frac{50}{5}

Divide the numbers:

\frac{50}{5} = 10

= 10

\frac{\frac{π}{2}}{5} = \frac{π}{10}

\frac{\frac{π}{2}}{5}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{π}{10}

= 10 + \frac{2 πn}{5} + \frac{π}{10}

x = 10 + \frac{2 πn}{5} + \frac{π}{10}

x = 10 + \frac{2 πn}{5} + \frac{π}{10}

x = 10 + \frac{2 πn}{5} + \frac{π}{10}

Solve

- 10 + 3 x - 40 + 2 x = \frac{3 π}{2} + 2 πn : x = 10 + \frac{2 πn}{5} + \frac{3 π}{10}

- 10 + 3 x - 40 + 2 x = \frac{3 π}{2} + 2 πn

Group like terms

3 x + 2 x - 10 - 40 = \frac{3 π}{2} + 2 πn

Add similar elements:

3 x + 2 x = 5 x

5 x - 10 - 40 = \frac{3 π}{2} + 2 πn

Subtract the numbers:

- 10 - 40 = - 50

5 x - 50 = \frac{3 π}{2} + 2 πn

Move

50

to the right side

5 x - 50 = \frac{3 π}{2} + 2 πn

Add

50

to both sides

5 x - 50 + 50 = \frac{3 π}{2} + 2 πn + 50

Simplify

5 x = \frac{3 π}{2} + 2 πn + 50

5 x = \frac{3 π}{2} + 2 πn + 50

Divide both sides by

5

5 x = \frac{3 π}{2} + 2 πn + 50

Divide both sides by

5

\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5} + \frac{50}{5}

Simplify

\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5} + \frac{50}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5} + \frac{50}{5} : 10 + \frac{2 πn}{5} + \frac{3 π}{10}

\frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5} + \frac{50}{5}

Group like terms

= \frac{50}{5} + \frac{2 πn}{5} + \frac{\frac{3 π}{2}}{5}

\frac{50}{5} = 10

\frac{50}{5}

Divide the numbers:

\frac{50}{5} = 10

= 10

\frac{\frac{3 π}{2}}{5} = \frac{3 π}{10}

\frac{\frac{3 π}{2}}{5}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{3 π}{10}

= 10 + \frac{2 πn}{5} + \frac{3 π}{10}

x = 10 + \frac{2 πn}{5} + \frac{3 π}{10}

x = 10 + \frac{2 πn}{5} + \frac{3 π}{10}

x = 10 + \frac{2 πn}{5} + \frac{3 π}{10}

x = 10 + \frac{2 πn}{5} + \frac{π}{10}, x = 10 + \frac{2 πn}{5} + \frac{3 π}{10}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for tan(3x-10)=cot(2x-40) ?
The general solution for tan(3x-10)=cot(2x-40) is x=10+(2pin)/5+pi/(10),x=10+(2pin)/5+(3pi)/(10)