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

The general solution for sec^2(x)-4sec(x)=0 is x=1.31811…+2pin,x=2pi-1.31811…+2pin

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Popular >

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

Solution

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

Solution

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

Degrees

x = 75.52248 \dots^{\circ} + 36 0^{\circ} n, x = 284.47751 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sec (x) = u

u^{2} - 4 u = 0

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

u^{2} - 4 u = 0

Solve with the quadratic formula

u^{2} - 4 u = 0

Quadratic Equation Formula:

For

a = 1, b = - 4, c = 0

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

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

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

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

Apply exponent rule:

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

n

is even

(- 4)^{2} = 4^{2}

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

Apply rule

0 \cdot a = 0

= 4^{2} - 0

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

= 4^{2}

Apply radical rule:

assuming

a \geq 0

= 4

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

Separate the solutions

u_{1} = \frac{- ( - 4 ) + 4}{2 \cdot 1}, u_{2} = \frac{- ( - 4 ) - 4}{2 \cdot 1}

u = \frac{- ( - 4 ) + 4}{2 \cdot 1} : 4

\frac{- ( - 4 ) + 4}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{4 + 4}{2 \cdot 1}

Add the numbers:

4 + 4 = 8

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4

u = \frac{- ( - 4 ) - 4}{2 \cdot 1} : 0

\frac{- ( - 4 ) - 4}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{4 - 4}{2 \cdot 1}

Subtract the numbers:

4 - 4 = 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 = 4, u = 0

Substitute back

u = sec (x)

sec (x) = 4, sec (x) = 0

sec (x) = 4, sec (x) = 0

sec (x) = 4 : x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

sec (x) = 4

Apply trig inverse properties

sec (x) = 4

General solutions for

sec (x) = 4

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

x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 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 (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

Show solutions in decimal form

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

Popular Examples

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sin (x) = \frac{5}{8}

sin(x)= 5/7

sin (x) = \frac{5}{7}

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sec (t) = 2

2cos^2(x)=-cos(x)

2 cos^{2} (x) = - cos (x)

2tan^2(x)+5tan(x)+3=0

2 tan^{2} (x) + 5 tan (x) + 3 = 0

Frequently Asked Questions (FAQ)

What is the general solution for sec^2(x)-4sec(x)=0 ?
The general solution for sec^2(x)-4sec(x)=0 is x=1.31811…+2pin,x=2pi-1.31811…+2pin