What is the general solution for (1+tan(x/2))/(1-tan(x/2))=sqrt(4.137131) ?

The general solution for (1+tan(x/2))/(1-tan(x/2))=sqrt(4.137131) is x=2*0.32845…+2pin

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(1+tan(x/2))/(1-tan(x/2))=sqrt(4.137131)

Solution

\frac{1 + tan ( \frac{x}{2} )}{1 - tan ( \frac{x}{2} )} = 4.137131

Solution

x = 2 \cdot 0.32845 \dots + 2 πn

Degrees

x = 37.63851 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{1 + tan ( \frac{x}{2} )}{1 - tan ( \frac{x}{2} )} = 4.137131

Solve by substitution

\frac{1 + tan ( \frac{x}{2} )}{1 - tan ( \frac{x}{2} )} = 4.137131

Let:

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

\frac{1 + u}{1 - u} = 4.137131

\frac{1 + u}{1 - u} = 4.137131 : u = \frac{5.137131 - 2 4.137131}{3.137131}

\frac{1 + u}{1 - u} = 4.137131

Multiply both sides by

1 - u

\frac{1 + u}{1 - u} = 4.137131

Multiply both sides by

1 - u

\frac{1 + u}{1 - u} (1 - u) = 4.137131 (1 - u)

Simplify

1 + u = 4.137131 (1 - u)

1 + u = 4.137131 (1 - u)

Expand

4.137131 (1 - u) : 4.137131 - 4.137131 u

4.137131 (1 - u)

Apply the distributive law:

a (b - c) = ab - a c

a = 4.137131, b = 1, c = u

= 4.137131 \cdot 1 - 4.137131 u

= 1 \cdot 4.137131 - 4.137131 u

Multiply:

1 \cdot 4.137131 = 4.137131

= 4.137131 - 4.137131 u

1 + u = 4.137131 - 4.137131 u

Move

1

to the right side

1 + u = 4.137131 - 4.137131 u

Subtract

1

from both sides

1 + u - 1 = 4.137131 - 4.137131 u - 1

Simplify

u = 4.137131 - 4.137131 u - 1

u = 4.137131 - 4.137131 u - 1

Move

4.137131 u

to the left side

u = 4.137131 - 4.137131 u - 1

Add

4.137131 u

to both sides

u + 4.137131 u = 4.137131 - 4.137131 u - 1 + 4.137131 u

Simplify

u + 4.137131 u = 4.137131 - 1

u + 4.137131 u = 4.137131 - 1

Factor

u + 4.137131 u : (1 + 4.137131) u

u + 4.137131 u

Factor out common term

u

= u (1 + 4.137131)

(1 + 4.137131) u = 4.137131 - 1

Divide both sides by

1 + 4.137131

(1 + 4.137131) u = 4.137131 - 1

Divide both sides by

1 + 4.137131

\frac{( 1 + 4.137131 ) u}{1 + 4.137131} = \frac{4.137131}{1 + 4.137131} - \frac{1}{1 + 4.137131}

Simplify

\frac{( 1 + 4.137131 ) u}{1 + 4.137131} = \frac{4.137131}{1 + 4.137131} - \frac{1}{1 + 4.137131}

Simplify

\frac{( 1 + 4.137131 ) u}{1 + 4.137131} : u

\frac{( 1 + 4.137131 ) u}{1 + 4.137131}

Cancel the common factor:

1 + 4.137131

= u

Simplify

\frac{4.137131}{1 + 4.137131} - \frac{1}{1 + 4.137131} : \frac{5.137131 - 2 4.137131}{3.137131}

\frac{4.137131}{1 + 4.137131} - \frac{1}{1 + 4.137131}

Apply rule

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

= \frac{4.137131 - 1}{1 + 4.137131}

Multiply by the conjugate

\frac{1 - 4.137131}{1 - 4.137131}

= \frac{( 4.137131 - 1 ) ( 1 - 4.137131 )}{( 1 + 4.137131 ) ( 1 - 4.137131 )}

(4.137131 - 1) (1 - 4.137131) = 2 4.137131 - 5.137131

(4.137131 - 1) (1 - 4.137131)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 4.137131, b = - 1, c = 1, d = - 4.137131

= 4.137131 \cdot 1 + 4.137131 (- 4.137131) + (- 1) \cdot 1 + (- 1) (- 4.137131)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= 1 \cdot 4.137131 - 4.137131 4.137131 - 1 \cdot 1 + 1 \cdot 4.137131

Simplify

1 \cdot 4.137131 - 4.137131 4.137131 - 1 \cdot 1 + 1 \cdot 4.137131 : 2 4.137131 - 5.137131

1 \cdot 4.137131 - 4.137131 4.137131 - 1 \cdot 1 + 1 \cdot 4.137131

Add similar elements:

1 \cdot 4.137131 + 1 \cdot 4.137131 = 2 4.137131

= 2 4.137131 - 4.137131 4.137131 - 1 \cdot 1

Apply radical rule:

a a = a

4.137131 4.137131 = 4.137131

= 2 4.137131 - 4.137131 - 1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 2 4.137131 - 4.137131 - 1

Subtract the numbers:

- 4.137131 - 1 = - 5.137131

= 2 4.137131 - 5.137131

= 2 4.137131 - 5.137131

(1 + 4.137131) (1 - 4.137131) = - 3.137131

(1 + 4.137131) (1 - 4.137131)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = 4.137131

= 1^{2} - (4.137131)^{2}

Simplify

1^{2} - (4.137131)^{2} : - 3.137131

1^{2} - (4.137131)^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - (4.137131)^{2}

(4.137131)^{2} = 4.137131

(4.137131)^{2}

Apply radical rule:

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

= (4.13713 1^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 4.13713 1^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 4.137131

= 1 - 4.137131

Subtract the numbers:

1 - 4.137131 = - 3.137131

= - 3.137131

= - 3.137131

= \frac{2 4.137131 - 5.137131}{- 3.137131}

Apply the fraction rule:

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

2 4.137131 - 5.137131 = - (5.137131 - 2 4.137131)

= \frac{5.137131 - 2 4.137131}{3.137131}

u = \frac{5.137131 - 2 4.137131}{3.137131}

u = \frac{5.137131 - 2 4.137131}{3.137131}

u = \frac{5.137131 - 2 4.137131}{3.137131}

Verify Solutions

Find undefined (singularity) points:

u = 1

Take the denominator(s) of

\frac{1 + u}{1 - u}

and compare to zero

Solve

1 - u = 0 : u = 1

1 - u = 0

Move

1

to the right side

1 - u = 0

Subtract

1

from both sides

1 - u - 1 = 0 - 1

Simplify

- u = - 1

- u = - 1

Divide both sides by

- 1

- u = - 1

Divide both sides by

- 1

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

Simplify

u = 1

u = 1

The following points are undefined

u = 1

Combine undefined points with solutions:

u = \frac{5.137131 - 2 4.137131}{3.137131}

Substitute back

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

tan (\frac{x}{2}) = \frac{5.137131 - 2 4.137131}{3.137131}

tan (\frac{x}{2}) = \frac{5.137131 - 2 4.137131}{3.137131}

tan (\frac{x}{2}) = \frac{5.137131 - 2 4.137131}{3.137131} : x = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

tan (\frac{x}{2}) = \frac{5.137131 - 2 4.137131}{3.137131}

Apply trig inverse properties

tan (\frac{x}{2}) = \frac{5.137131 - 2 4.137131}{3.137131}

General solutions for

tan (\frac{x}{2}) = \frac{5.137131 - 2 4.137131}{3.137131}

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

\frac{x}{2} = arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + πn

\frac{x}{2} = arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + πn

Solve

\frac{x}{2} = arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + πn : x = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

\frac{x}{2} = arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + πn

Multiply both sides by

2

\frac{x}{2} = arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + πn

Multiply both sides by

2

\frac{2 x}{2} = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

Simplify

x = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

x = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

x = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

Combine all the solutions

x = 2 arctan (\frac{5.137131 - 2 4.137131}{3.137131}) + 2 πn

Show solutions in decimal form

x = 2 \cdot 0.32845 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for (1+tan(x/2))/(1-tan(x/2))=sqrt(4.137131) ?
The general solution for (1+tan(x/2))/(1-tan(x/2))=sqrt(4.137131) is x=2*0.32845…+2pin