What is the general solution for -sec^2(x)+1=-5sec(x)-5 ?

The general solution for -sec^2(x)+1=-5sec(x)-5 is x=pi+2pin,x=1.40334…+2pin,x=2pi-1.40334…+2pin

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-sec^2(x)+1=-5sec(x)-5

Solution

- sec^{2} (x) + 1 = - 5 sec (x) - 5

Solution

x = π + 2 πn, x = 1.40334 \dots + 2 πn, x = 2 π - 1.40334 \dots + 2 πn

Degrees

x = 18 0^{\circ} + 36 0^{\circ} n, x = 80.40593 \dots^{\circ} + 36 0^{\circ} n, x = 279.59406 \dots^{\circ} + 36 0^{\circ} n

Solution steps

- sec^{2} (x) + 1 = - 5 sec (x) - 5

Solve by substitution

- sec^{2} (x) + 1 = - 5 sec (x) - 5

Let:

sec (x) = u

- u^{2} + 1 = - 5 u - 5

- u^{2} + 1 = - 5 u - 5 : u = - 1, u = 6

- u^{2} + 1 = - 5 u - 5

Move

5

to the left side

- u^{2} + 1 = - 5 u - 5

Add

5

to both sides

- u^{2} + 1 + 5 = - 5 u - 5 + 5

Simplify

- u^{2} + 6 = - 5 u

- u^{2} + 6 = - 5 u

Move

5 u

to the left side

- u^{2} + 6 = - 5 u

Add

5 u

to both sides

- u^{2} + 6 + 5 u = - 5 u + 5 u

Simplify

- u^{2} + 6 + 5 u = 0

- u^{2} + 6 + 5 u = 0

Write in the standard form

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

- u^{2} + 5 u + 6 = 0

Solve with the quadratic formula

- u^{2} + 5 u + 6 = 0

Quadratic Equation Formula:

For

a = - 1, b = 5, c = 6

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 1 ) \cdot 6}{2 ( - 1 )}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 1 ) \cdot 6}{2 ( - 1 )}

5^{2} - 4 (- 1) \cdot 6 = 7

5^{2} - 4 (- 1) \cdot 6

Apply rule

- (- a) = a

= 5^{2} + 4 \cdot 1 \cdot 6

Multiply the numbers:

4 \cdot 1 \cdot 6 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

Add the numbers:

25 + 24 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- 5 \pm 7}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- 5 + 7}{2 ( - 1 )}, u_{2} = \frac{- 5 - 7}{2 ( - 1 )}

u = \frac{- 5 + 7}{2 ( - 1 )} : - 1

\frac{- 5 + 7}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 5 + 7}{- 2 \cdot 1}

Add/Subtract the numbers:

- 5 + 7 = 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

u = \frac{- 5 - 7}{2 ( - 1 )} : 6

\frac{- 5 - 7}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 5 - 7}{- 2 \cdot 1}

Subtract the numbers:

- 5 - 7 = - 12

= \frac{- 12}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 12}{- 2}

Apply the fraction rule:

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

= \frac{12}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6

The solutions to the quadratic equation are:

u = - 1, u = 6

Substitute back

u = sec (x)

sec (x) = - 1, sec (x) = 6

sec (x) = - 1, sec (x) = 6

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) = 6 : x = arcsec (6) + 2 πn, x = 2 π - arcsec (6) + 2 πn

sec (x) = 6

Apply trig inverse properties

sec (x) = 6

General solutions for

sec (x) = 6

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

x = arcsec (6) + 2 πn, x = 2 π - arcsec (6) + 2 πn

x = arcsec (6) + 2 πn, x = 2 π - arcsec (6) + 2 πn

Combine all the solutions

x = π + 2 πn, x = arcsec (6) + 2 πn, x = 2 π - arcsec (6) + 2 πn

Show solutions in decimal form

x = π + 2 πn, x = 1.40334 \dots + 2 πn, x = 2 π - 1.40334 \dots + 2 πn

Popular Examples

sec(t)= 1/(-(sqrt(2))/2)

sec (t) = \frac{1}{- \frac{2}{2}}

sin(x)=0.556692

sin (x) = 0.556692

tan^2(x)+4tan(x)+3=0

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

(1+tan(x))^2+(1-tan(x))^2=sec^2(x)

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

tan(x)=-(sqrt(2))/2

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

Frequently Asked Questions (FAQ)

What is the general solution for -sec^2(x)+1=-5sec(x)-5 ?
The general solution for -sec^2(x)+1=-5sec(x)-5 is x=pi+2pin,x=1.40334…+2pin,x=2pi-1.40334…+2pin