What is the general solution for sec^3(x)=-1 ?

The general solution for sec^3(x)=-1 is x=pi+2pin

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sec^3(x)=-1

Solution

sec^{3} (x) = - 1

Solution

x = π + 2 πn

Degrees

x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

sec^{3} (x) = - 1

Solve by substitution

sec^{3} (x) = - 1

Let:

sec (x) = u

u^{3} = - 1

u^{3} = - 1 : u = - 1, u = \frac{1}{2} - i \frac{3}{2}, u = \frac{1}{2} + i \frac{3}{2}

u^{3} = - 1

For

x^{3} = f (a)

the solutions are

n

is odd

= - 1

Simplify

n

is odd

= - 1

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

Multiply:

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

= - \frac{- 1 + 3 i}{2}

Rewrite

- \frac{- 1 + 3 i}{2}

in standard complex form:

\frac{1}{2} - \frac{3}{2} i

- \frac{- 1 + 3 i}{2}

Apply the fraction rule:

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

\frac{- 1 + 3 i}{2} = - (- \frac{1}{2}) - (\frac{3 i}{2})

= - (- \frac{1}{2}) - (\frac{3 i}{2})

Remove parentheses:

(a) = a, - (- a) = a

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

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

Simplify

n

is odd

= - 1

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

Multiply:

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

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

Rewrite

- \frac{- 1 - 3 i}{2}

in standard complex form:

\frac{1}{2} + \frac{3}{2} i

- \frac{- 1 - 3 i}{2}

Apply the fraction rule:

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

\frac{- 1 - 3 i}{2} = - (- \frac{1}{2}) - (- \frac{3 i}{2})

= - (- \frac{1}{2}) - (- \frac{3 i}{2})

Apply rule

- (- a) = a

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

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

u = - 1, u = \frac{1}{2} - i \frac{3}{2}, u = \frac{1}{2} + i \frac{3}{2}

Substitute back

u = sec (x)

sec (x) = - 1, sec (x) = \frac{1}{2} - i \frac{3}{2}, sec (x) = \frac{1}{2} + i \frac{3}{2}

sec (x) = - 1, sec (x) = \frac{1}{2} - i \frac{3}{2}, sec (x) = \frac{1}{2} + i \frac{3}{2}

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

General solutions for

sec (x) = - 1

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 = π + 2 πn

x = π + 2 πn

sec (x) = \frac{1}{2} - i \frac{3}{2} :

No Solution

sec (x) = \frac{1}{2} - i \frac{3}{2}

No Solution

sec (x) = \frac{1}{2} + i \frac{3}{2} :

No Solution

sec (x) = \frac{1}{2} + i \frac{3}{2}

No Solution

Combine all the solutions

x = π + 2 πn

Popular Examples

1/(cot^2(x))+sqrt(3)tan(x)=0

\frac{1}{cot ^{2} ( x )} + 3 tan (x) = 0

tan(x)=0.23

tan (x) = 0.23

tan(x)=0.45

tan (x) = 0.45

2cos(x)-1=sec(x)

2 cos (x) - 1 = sec (x)

sqrt(2)-cos(x)=sqrt(2)+sin(x)

2 - cos (x) = 2 + sin (x)

Frequently Asked Questions (FAQ)

What is the general solution for sec^3(x)=-1 ?
The general solution for sec^3(x)=-1 is x=pi+2pin