What is the general solution for 2tan^2(x)+4sec(x)+sec^2(x)=2 ?

The general solution for 2tan^2(x)+4sec(x)+sec^2(x)=2 is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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2tan^2(x)+4sec(x)+sec^2(x)=2

Solution

2 tan^{2} (x) + 4 sec (x) + sec^{2} (x) = 2

Solution

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

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

2 tan^{2} (x) + 4 sec (x) + sec^{2} (x) = 2

Subtract

2

from both sides

2 tan^{2} (x) + 4 sec (x) + sec^{2} (x) - 2 = 0

Rewrite using trig identities

- 2 + sec^{2} (x) + 2 tan^{2} (x) + 4 sec (x)

Use the Pythagorean identity:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= - 2 + sec^{2} (x) + 2 (sec^{2} (x) - 1) + 4 sec (x)

Simplify

- 2 + sec^{2} (x) + 2 (sec^{2} (x) - 1) + 4 sec (x) : 3 sec^{2} (x) + 4 sec (x) - 4

- 2 + sec^{2} (x) + 2 (sec^{2} (x) - 1) + 4 sec (x)

Expand

2 (sec^{2} (x) - 1) : 2 sec^{2} (x) - 2

2 (sec^{2} (x) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = sec^{2} (x), c = 1

= 2 sec^{2} (x) - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 sec^{2} (x) - 2

= - 2 + sec^{2} (x) + 2 sec^{2} (x) - 2 + 4 sec (x)

Simplify

- 2 + sec^{2} (x) + 2 sec^{2} (x) - 2 + 4 sec (x) : 3 sec^{2} (x) + 4 sec (x) - 4

- 2 + sec^{2} (x) + 2 sec^{2} (x) - 2 + 4 sec (x)

Add similar elements:

sec^{2} (x) + 2 sec^{2} (x) = 3 sec^{2} (x)

= - 2 + 3 sec^{2} (x) - 2 + 4 sec (x)

Group like terms

= 3 sec^{2} (x) + 4 sec (x) - 2 - 2

Subtract the numbers:

- 2 - 2 = - 4

= 3 sec^{2} (x) + 4 sec (x) - 4

= 3 sec^{2} (x) + 4 sec (x) - 4

= 3 sec^{2} (x) + 4 sec (x) - 4

- 4 + 3 sec^{2} (x) + 4 sec (x) = 0

Solve by substitution

- 4 + 3 sec^{2} (x) + 4 sec (x) = 0

Let:

sec (x) = u

- 4 + 3 u^{2} + 4 u = 0

- 4 + 3 u^{2} + 4 u = 0 : u = \frac{2}{3}, u = - 2

- 4 + 3 u^{2} + 4 u = 0

Write in the standard form

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

3 u^{2} + 4 u - 4 = 0

Solve with the quadratic formula

3 u^{2} + 4 u - 4 = 0

Quadratic Equation Formula:

For

a = 3, b = 4, c = - 4

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

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

4^{2} - 4 \cdot 3 (- 4) = 8

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

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 3 \cdot 4

Multiply the numbers:

4 \cdot 3 \cdot 4 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

Add the numbers:

16 + 48 = 64

= 64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

8^{2} = 8

= 8

u_{1, 2} = \frac{- 4 \pm 8}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- 4 + 8}{2 \cdot 3}, u_{2} = \frac{- 4 - 8}{2 \cdot 3}

u = \frac{- 4 + 8}{2 \cdot 3} : \frac{2}{3}

\frac{- 4 + 8}{2 \cdot 3}

Add/Subtract the numbers:

- 4 + 8 = 4

= \frac{4}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{4}{6}

Cancel the common factor:

2

= \frac{2}{3}

u = \frac{- 4 - 8}{2 \cdot 3} : - 2

\frac{- 4 - 8}{2 \cdot 3}

Subtract the numbers:

- 4 - 8 = - 12

= \frac{- 12}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 12}{6}

Apply the fraction rule:

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

= - \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= - 2

The solutions to the quadratic equation are:

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

Substitute back

u = sec (x)

sec (x) = \frac{2}{3}, sec (x) = - 2

sec (x) = \frac{2}{3}, sec (x) = - 2

sec (x) = \frac{2}{3} :

No Solution

sec (x) = \frac{2}{3}

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

sec (x) = - 2 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

sec (x) = - 2

General solutions for

sec (x) = - 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

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

Combine all the solutions

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

Popular Examples

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2 + cos (2 x) = - 3 cos (x), 0 \leq x \leq 2 π

sec(x+10)=5.759

sec (x + 1 0^{\circ}) = 5.759

cos(3x+pi/4)=(sqrt(3))/2

cos (3 x + \frac{π}{4}) = \frac{3}{2}

-sin(4x)=sin(8x)

- sin (4 x) = sin (8 x)

tan(θ)=(3sqrt(3))/3

tan (θ) = \frac{3 3}{3}

Frequently Asked Questions (FAQ)

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