What is the general solution for 4-sec^2(x)=0 ?

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

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4-sec^2(x)=0

Solution

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

Solution

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sec (x) = u

4 - u^{2} = 0

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

4 - u^{2} = 0

Move

4

to the right side

4 - u^{2} = 0

Subtract

4

from both sides

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

Simplify

- u^{2} = - 4

- u^{2} = - 4

Divide both sides by

- 1

- u^{2} = - 4

Divide both sides by

- 1

\frac{- u ^{2}}{- 1} = \frac{- 4}{- 1}

Simplify

u^{2} = 4

u^{2} = 4

For

x^{2} = f (a)

the solutions are

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

u = 4, u = - 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= - 2

u = 2, u = - 2

Substitute back

u = sec (x)

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

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

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

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

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

Popular Examples

3tan^2(x)-9=0

3 tan^{2} (x) - 9 = 0

3cos(4x)=2

3 cos (4 x) = 2

2cos^2(x)=cos(x)+1

2 cos^{2} (x) = cos (x) + 1

4cos(θ)+2=0

4 cos (θ) + 2 = 0

6cos^2(θ)=3

6 cos^{2} (θ) = 3

Frequently Asked Questions (FAQ)

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