What is the general solution for tan^2(x/2)-tan(x/2)-2=0 ?

The general solution for tan^2(x/2)-tan(x/2)-2=0 is x=2*1.10714…+2pin,x=(3pi)/2+2pin

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tan^2(x/2)-tan(x/2)-2=0

Solution

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

Solution

x = 2 \cdot 1.10714 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 126.86989 \dots^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

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

u^{2} - u - 2 = 0

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

u^{2} - u - 2 = 0

Solve with the quadratic formula

u^{2} - u - 2 = 0

Quadratic Equation Formula:

For

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

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

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

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

(- 1)^{2} - 4 \cdot 1 \cdot (- 2)

Apply rule

- (- a) = a

= (- 1)^{2} + 4 \cdot 1 \cdot 2

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 1 \cdot 2 = 8

4 \cdot 1 \cdot 2

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 8

= 1 + 8

Add the numbers:

1 + 8 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

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

Separate the solutions

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

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

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

Apply rule

- (- a) = a

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

Add the numbers:

1 + 3 = 4

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

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

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

Apply rule

- (- a) = a

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

Subtract the numbers:

1 - 3 = - 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

The solutions to the quadratic equation are:

u = 2, u = - 1

Substitute back

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

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

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

tan (\frac{x}{2}) = 2 : x = 2 arctan (2) + 2 πn

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

Apply trig inverse properties

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

General solutions for

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

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

\frac{x}{2} = arctan (2) + πn

\frac{x}{2} = arctan (2) + πn

Solve

\frac{x}{2} = arctan (2) + πn : x = 2 arctan (2) + 2 πn

\frac{x}{2} = arctan (2) + πn

Multiply both sides by

2

\frac{x}{2} = arctan (2) + πn

Multiply both sides by

2

\frac{2 x}{2} = 2 arctan (2) + 2 πn

Simplify

x = 2 arctan (2) + 2 πn

x = 2 arctan (2) + 2 πn

x = 2 arctan (2) + 2 πn

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

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

General solutions for

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

\frac{x}{2} = \frac{3 π}{4} + πn

\frac{x}{2} = \frac{3 π}{4} + πn

Solve

\frac{x}{2} = \frac{3 π}{4} + πn : x = \frac{3 π}{2} + 2 πn

\frac{x}{2} = \frac{3 π}{4} + πn

Multiply both sides by

2

\frac{x}{2} = \frac{3 π}{4} + πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{3 π}{4} + 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{3 π}{4} + 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{3 π}{4} + 2 πn : \frac{3 π}{2} + 2 πn

2 \cdot \frac{3 π}{4} + 2 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

Multiply fractions:

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

= \frac{3 π 2}{4}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = 2 arctan (2) + 2 πn, x = \frac{3 π}{2} + 2 πn

Show solutions in decimal form

x = 2 \cdot 1.10714 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x/2)-tan(x/2)-2=0 ?
The general solution for tan^2(x/2)-tan(x/2)-2=0 is x=2*1.10714…+2pin,x=(3pi)/2+2pin