What is the general solution for sec^2(x)-8sec(x)=0 ?

The general solution for sec^2(x)-8sec(x)=0 is x=1.44546…+2pin,x=2pi-1.44546…+2pin

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Popular >

sec^2(x)-8sec(x)=0

Solution

sec^{2} (x) - 8 sec (x) = 0

Solution

x = 1.44546 \dots + 2 πn, x = 2 π - 1.44546 \dots + 2 πn

Degrees

x = 82.81924 \dots^{\circ} + 36 0^{\circ} n, x = 277.18075 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sec^{2} (x) - 8 sec (x) = 0

Solve by substitution

sec^{2} (x) - 8 sec (x) = 0

Let:

sec (x) = u

u^{2} - 8 u = 0

u^{2} - 8 u = 0 : u = 8, u = 0

u^{2} - 8 u = 0

Solve with the quadratic formula

u^{2} - 8 u = 0

Quadratic Equation Formula:

For

a = 1, b = - 8, c = 0

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 8)^{2} - 4 \cdot 1 \cdot 0 = 8

(- 8)^{2} - 4 \cdot 1 \cdot 0

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 8)^{2} = 8^{2}

= 8^{2} - 4 \cdot 1 \cdot 0

Apply rule

0 \cdot a = 0

= 8^{2} - 0

8^{2} - 0 = 8^{2}

= 8^{2}

Apply radical rule:

assuming

a \geq 0

= 8

u_{1, 2} = \frac{- ( - 8 ) \pm 8}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 8 ) + 8}{2 \cdot 1}, u_{2} = \frac{- ( - 8 ) - 8}{2 \cdot 1}

u = \frac{- ( - 8 ) + 8}{2 \cdot 1} : 8

\frac{- ( - 8 ) + 8}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{8 + 8}{2 \cdot 1}

Add the numbers:

8 + 8 = 16

= \frac{16}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{16}{2}

Divide the numbers:

\frac{16}{2} = 8

= 8

u = \frac{- ( - 8 ) - 8}{2 \cdot 1} : 0

\frac{- ( - 8 ) - 8}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{8 - 8}{2 \cdot 1}

Subtract the numbers:

8 - 8 = 0

= \frac{0}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = 8, u = 0

Substitute back

u = sec (x)

sec (x) = 8, sec (x) = 0

sec (x) = 8, sec (x) = 0

sec (x) = 8 : x = arcsec (8) + 2 πn, x = 2 π - arcsec (8) + 2 πn

sec (x) = 8

Apply trig inverse properties

sec (x) = 8

General solutions for

sec (x) = 8

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

x = arcsec (8) + 2 πn, x = 2 π - arcsec (8) + 2 πn

x = arcsec (8) + 2 πn, x = 2 π - arcsec (8) + 2 πn

sec (x) = 0 :

No Solution

sec (x) = 0

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

No Solution

Combine all the solutions

x = arcsec (8) + 2 πn, x = 2 π - arcsec (8) + 2 πn

Show solutions in decimal form

x = 1.44546 \dots + 2 πn, x = 2 π - 1.44546 \dots + 2 πn

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7 tan (A) + 42 = 0

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2 sin (x + \frac{π}{4}) = 1

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sin(x+pi/4)=-(sqrt(2))/2 ,0<= x<= 2pi

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Frequently Asked Questions (FAQ)

What is the general solution for sec^2(x)-8sec(x)=0 ?
The general solution for sec^2(x)-8sec(x)=0 is x=1.44546…+2pin,x=2pi-1.44546…+2pin