What is the general solution for 2/(tan(x))=3-tan(x) ?

The general solution for 2/(tan(x))=3-tan(x) is x= pi/4+pin,x=1.10714…+pin

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2/(tan(x))=3-tan(x)

Solution

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

Solution

x = \frac{π}{4} + πn, x = 1.10714 \dots + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

\frac{2}{u} u = 3 u - uu

Simplify

- uu : - u^{2}

\frac{2}{u} u = 3 u - uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= - u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - u^{2}

2 = 3 u - u^{2}

2 = 3 u - u^{2}

Solve

2 = 3 u - u^{2} : u = 1, u = 2

2 = 3 u - u^{2}

Switch sides

3 u - u^{2} = 2

Move

2

to the left side

3 u - u^{2} = 2

Subtract

2

from both sides

3 u - u^{2} - 2 = 2 - 2

Simplify

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

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

Write in the standard form

a x^{2} + b x + c = 0

- u^{2} + 3 u - 2 = 0

Solve with the quadratic formula

- u^{2} + 3 u - 2 = 0

Quadratic Equation Formula:

For

a = - 1, b = 3, c = - 2

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 1 ) ( - 2 )}{2 ( - 1 )}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 1 ) ( - 2 )}{2 ( - 1 )}

3^{2} - 4 (- 1) (- 2) = 1

3^{2} - 4 (- 1) (- 2)

Apply rule

- (- a) = a

= 3^{2} - 4 \cdot 1 \cdot 2

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 3^{2} - 8

3^{2} = 9

= 9 - 8

Subtract the numbers:

9 - 8 = 1

= 1

Apply rule

1 = 1

= 1

u_{1, 2} = \frac{- 3 \pm 1}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- 3 + 1}{2 ( - 1 )}, u_{2} = \frac{- 3 - 1}{2 ( - 1 )}

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

\frac{- 3 + 1}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

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

Add/Subtract the numbers:

- 3 + 1 = - 2

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2}{- 2}

Apply the fraction rule:

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

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

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

\frac{- 3 - 1}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

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

Subtract the numbers:

- 3 - 1 = - 4

= \frac{- 4}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 4}{- 2}

Apply the fraction rule:

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

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

The solutions to the quadratic equation are:

u = 1, u = 2

u = 1, u = 2

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{2}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = 2

Substitute back

u = tan (x)

tan (x) = 1, tan (x) = 2

tan (x) = 1, tan (x) = 2

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

tan (x) = 2 : x = arctan (2) + πn

tan (x) = 2

Apply trig inverse properties

tan (x) = 2

General solutions for

tan (x) = 2

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

x = arctan (2) + πn

x = arctan (2) + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = arctan (2) + πn

Show solutions in decimal form

x = \frac{π}{4} + πn, x = 1.10714 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2/(tan(x))=3-tan(x) ?
The general solution for 2/(tan(x))=3-tan(x) is x= pi/4+pin,x=1.10714…+pin