What is the general solution for cot(x)*tan(2x)=3 ?

The general solution for cot(x)*tan(2x)=3 is x=0.52359…+pin,x=-0.52359…+pin

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cot(x)*tan(2x)=3

Solution

cot (x) \cdot tan (2 x) = 3

Solution

x = 0.52359 \dots + πn, x = - 0.52359 \dots + πn

Degrees

x = 3 0^{\circ} + 18 0^{\circ} n, x = - 3 0^{\circ} + 18 0^{\circ} n

Solution steps

cot (x) tan (2 x) = 3

Subtract

3

from both sides

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

cot (x) = \frac{1}{tan ( x )}

= - 3 + \frac{1}{tan ( x )} tan (2 x)

\frac{1}{tan ( x )} tan (2 x) = \frac{tan ( 2 x )}{tan ( x )}

\frac{1}{tan ( x )} tan (2 x)

Multiply fractions:

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

= \frac{1 \cdot tan ( 2 x )}{tan ( x )}

Multiply:

1 \cdot tan (2 x) = tan (2 x)

= \frac{tan ( 2 x )}{tan ( x )}

= - 3 + \frac{tan ( 2 x )}{tan ( x )}

Use the Double Angle identity:

tan (2 x) = \frac{2 tan ( x )}{1 - tan ^{2} ( x )}

= - 3 + \frac{\frac{2 t a n ( x )}{1 - t a n ^{2} ( x )}}{tan ( x )}

\frac{\frac{2 t a n ( x )}{1 - t a n ^{2} ( x )}}{tan ( x )} = \frac{2}{1 - tan ^{2} ( x )}

\frac{\frac{2 t a n ( x )}{1 - t a n ^{2} ( x )}}{tan ( x )}

Apply the fraction rule:

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

= \frac{2 tan ( x )}{( 1 - tan ^{2} ( x ) ) tan ( x )}

Cancel the common factor:

tan (x)

= \frac{2}{1 - tan ^{2} ( x )}

= - 3 + \frac{2}{1 - tan ^{2} ( x )}

- 3 + \frac{2}{1 - tan ^{2} ( x )} = 0

Solve by substitution

- 3 + \frac{2}{1 - tan ^{2} ( x )} = 0

Let:

tan (x) = u

- 3 + \frac{2}{1 - u ^{2}} = 0

- 3 + \frac{2}{1 - u ^{2}} = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

- 3 + \frac{2}{1 - u ^{2}} = 0

Multiply both sides by

1 - u^{2}

- 3 + \frac{2}{1 - u ^{2}} = 0

Multiply both sides by

1 - u^{2}

- 3 (1 - u^{2}) + \frac{2}{1 - u ^{2}} (1 - u^{2}) = 0 \cdot (1 - u^{2})

Simplify

- 3 (1 - u^{2}) + \frac{2}{1 - u ^{2}} (1 - u^{2}) = 0 \cdot (1 - u^{2})

Simplify

\frac{2}{1 - u ^{2}} (1 - u^{2}) : 2

\frac{2}{1 - u ^{2}} (1 - u^{2})

Multiply fractions:

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

= \frac{2 ( 1 - u ^{2} )}{1 - u ^{2}}

Cancel the common factor:

1 - u^{2}

= 2

Simplify

0 \cdot (1 - u^{2}) : 0

0 \cdot (1 - u^{2})

Apply rule

0 \cdot a = 0

= 0

- 3 (1 - u^{2}) + 2 = 0

- 3 (1 - u^{2}) + 2 = 0

- 3 (1 - u^{2}) + 2 = 0

Solve

- 3 (1 - u^{2}) + 2 = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

- 3 (1 - u^{2}) + 2 = 0

Move

2

to the right side

- 3 (1 - u^{2}) + 2 = 0

Subtract

2

from both sides

- 3 (1 - u^{2}) + 2 - 2 = 0 - 2

Simplify

- 3 (1 - u^{2}) = - 2

- 3 (1 - u^{2}) = - 2

Divide both sides by

- 3

- 3 (1 - u^{2}) = - 2

Divide both sides by

- 3

\frac{- 3 ( 1 - u ^{2} )}{- 3} = \frac{- 2}{- 3}

Simplify

1 - u^{2} = \frac{2}{3}

1 - u^{2} = \frac{2}{3}

Move

1

to the right side

1 - u^{2} = \frac{2}{3}

Subtract

1

from both sides

1 - u^{2} - 1 = \frac{2}{3} - 1

Simplify

1 - u^{2} - 1 = \frac{2}{3} - 1

Simplify

1 - u^{2} - 1 : - u^{2}

1 - u^{2} - 1

Add similar elements:

1 - 1 = 0

= - u^{2}

Simplify

\frac{2}{3} - 1 : - \frac{1}{3}

\frac{2}{3} - 1

Convert element to fraction:

1 = \frac{1 \cdot 3}{3}

= - \frac{1 \cdot 3}{3} + \frac{2}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot 3 + 2}{3}

- 1 \cdot 3 + 2 = - 1

- 1 \cdot 3 + 2

Multiply the numbers:

1 \cdot 3 = 3

= - 3 + 2

Add/Subtract the numbers:

- 3 + 2 = - 1

= - 1

= \frac{- 1}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{3}

- u^{2} = - \frac{1}{3}

- u^{2} = - \frac{1}{3}

- u^{2} = - \frac{1}{3}

Divide both sides by

- 1

- u^{2} = - \frac{1}{3}

Divide both sides by

- 1

\frac{- u ^{2}}{- 1} = \frac{- \frac{1}{3}}{- 1}

Simplify

\frac{- u ^{2}}{- 1} = \frac{- \frac{1}{3}}{- 1}

Simplify

\frac{- u ^{2}}{- 1} : u^{2}

\frac{- u ^{2}}{- 1}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{u ^{2}}{1}

Apply rule

\frac{a}{1} = a

= u^{2}

Simplify

\frac{- \frac{1}{3}}{- 1} : \frac{1}{3}

\frac{- \frac{1}{3}}{- 1}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{\frac{1}{3}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= \frac{1}{3}

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{3}, u = - \frac{1}{3}

u = \frac{1}{3}, u = - \frac{1}{3}

Verify Solutions

Find undefined (singularity) points:

u = 1, u = - 1

Take the denominator(s) of

- 3 + \frac{2}{1 - u ^{2}}

and compare to zero

Solve

1 - u^{2} = 0 : u = 1, u = - 1

1 - u^{2} = 0

Move

1

to the right side

1 - u^{2} = 0

Subtract

1

from both sides

1 - u^{2} - 1 = 0 - 1

Simplify

- u^{2} = - 1

- u^{2} = - 1

Divide both sides by

- 1

- u^{2} = - 1

Divide both sides by

- 1

\frac{- u ^{2}}{- 1} = \frac{- 1}{- 1}

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Apply radical rule:

1 = 1

= 1

- 1 = - 1

- 1

Apply radical rule:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

The following points are undefined

u = 1, u = - 1

Combine undefined points with solutions:

u = \frac{1}{3}, u = - \frac{1}{3}

Substitute back

u = tan (x)

tan (x) = \frac{1}{3}, tan (x) = - \frac{1}{3}

tan (x) = \frac{1}{3}, tan (x) = - \frac{1}{3}

tan (x) = \frac{1}{3} : x = arctan (\frac{1}{3}) + πn

tan (x) = \frac{1}{3}

Apply trig inverse properties

tan (x) = \frac{1}{3}

General solutions for

tan (x) = \frac{1}{3}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{1}{3}) + πn

x = arctan (\frac{1}{3}) + πn

tan (x) = - \frac{1}{3} : x = arctan (- \frac{1}{3}) + πn

tan (x) = - \frac{1}{3}

Apply trig inverse properties

tan (x) = - \frac{1}{3}

General solutions for

tan (x) = - \frac{1}{3}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{1}{3}) + πn

x = arctan (- \frac{1}{3}) + πn

Combine all the solutions

x = arctan (\frac{1}{3}) + πn, x = arctan (- \frac{1}{3}) + πn

Show solutions in decimal form

x = 0.52359 \dots + πn, x = - 0.52359 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for cot(x)*tan(2x)=3 ?
The general solution for cot(x)*tan(2x)=3 is x=0.52359…+pin,x=-0.52359…+pin