What is the general solution for sec^4(x)-4=0 ?

The general solution for sec^4(x)-4=0 is x= pi/4+2pin,x=(7pi)/4+2pin,x=(3pi)/4+2pin,x=(5pi)/4+2pin

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Popular Trigonometry >

sec^4(x)-4=0

Solution

sec^{4} (x) - 4 = 0

Solution

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n

Solution steps

sec^{4} (x) - 4 = 0

Solve by substitution

sec^{4} (x) - 4 = 0

Let:

sec (x) = u

u^{4} - 4 = 0

u^{4} - 4 = 0 : u = 2, u = - 2, u = 2 i, u = - 2 i

u^{4} - 4 = 0

Move

4

to the right side

u^{4} - 4 = 0

Add

4

to both sides

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

Simplify

u^{4} = 4

u^{4} = 4

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

v^{2} = 4

Solve

v^{2} = 4 : v = 4, v = - 4

v^{2} = 4

For

(g (x))^{2} = f (a)

the solutions are

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

v = 4, v = - 4

v = 4, v = - 4

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 4 : u = 2, u = - 2

u^{2} = 4

Simplify

4 : 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

For

x^{2} = f (a)

the solutions are

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

u = 2, u = - 2

Solve

u^{2} = - 4 : u = 2 i, u = - 2 i

u^{2} = - 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

u^{2} = - 2

For

x^{2} = f (a)

the solutions are

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

u = - 2, u = - - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

Simplify

- - 2 : - 2 i

- - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

The solutions are

u = 2, u = - 2, u = 2 i, u = - 2 i

Substitute back

u = sec (x)

sec (x) = 2, sec (x) = - 2, sec (x) = 2 i, sec (x) = - 2 i

sec (x) = 2, sec (x) = - 2, sec (x) = 2 i, sec (x) = - 2 i

sec (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 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{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) = - 2 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 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{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) = 2 i :

No Solution

sec (x) = 2 i

No Solution

sec (x) = - 2 i :

No Solution

sec (x) = - 2 i

No Solution

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sec^4(x)-4=0 ?
The general solution for sec^4(x)-4=0 is x= pi/4+2pin,x=(7pi)/4+2pin,x=(3pi)/4+2pin,x=(5pi)/4+2pin