What is the general solution for sec^2(x)-7sec(x)=0 ?

The general solution for sec^2(x)-7sec(x)=0 is x=1.42744…+2pin,x=2pi-1.42744…+2pin

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sec^2(x)-7sec(x)=0

Solution

sec^{2} (x) - 7 sec (x) = 0

Solution

x = 1.42744 \dots + 2 πn, x = 2 π - 1.42744 \dots + 2 πn

Degrees

x = 81.78678 \dots^{\circ} + 36 0^{\circ} n, x = 278.21321 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sec^{2} (x) - 7 sec (x) = 0

Solve by substitution

sec^{2} (x) - 7 sec (x) = 0

Let:

sec (x) = u

u^{2} - 7 u = 0

u^{2} - 7 u = 0 : u = 7, u = 0

u^{2} - 7 u = 0

Solve with the quadratic formula

u^{2} - 7 u = 0

Quadratic Equation Formula:

For

a = 1, b = - 7, c = 0

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 7)^{2} - 4 \cdot 1 \cdot 0 = 7

(- 7)^{2} - 4 \cdot 1 \cdot 0

Apply exponent rule:

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

n

is even

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 1 \cdot 0

Apply rule

0 \cdot a = 0

= 7^{2} - 0

7^{2} - 0 = 7^{2}

= 7^{2}

Apply radical rule:

assuming

a \geq 0

= 7

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

Separate the solutions

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

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

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

Apply rule

- (- a) = a

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

Add the numbers:

7 + 7 = 14

= \frac{14}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{14}{2}

Divide the numbers:

\frac{14}{2} = 7

= 7

u = \frac{- ( - 7 ) - 7}{2 \cdot 1} : 0

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

Apply rule

- (- a) = a

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

Subtract the numbers:

7 - 7 = 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 = 7, u = 0

Substitute back

u = sec (x)

sec (x) = 7, sec (x) = 0

sec (x) = 7, sec (x) = 0

sec (x) = 7 : x = arcsec (7) + 2 πn, x = 2 π - arcsec (7) + 2 πn

sec (x) = 7

Apply trig inverse properties

sec (x) = 7

General solutions for

sec (x) = 7

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

x = arcsec (7) + 2 πn, x = 2 π - arcsec (7) + 2 πn

x = arcsec (7) + 2 πn, x = 2 π - arcsec (7) + 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 (7) + 2 πn, x = 2 π - arcsec (7) + 2 πn

Show solutions in decimal form

x = 1.42744 \dots + 2 πn, x = 2 π - 1.42744 \dots + 2 πn

Popular Examples

e^{pi/2}-e^{arctan(1+x^2)}=0

e^{\frac{π}{2}} - e^{a r c t a n (1 + x^{2})} = 0

3cot(θ)-7=0

3 cot (θ) - 7 = 0

cos(x)= 1249/1250

cos (x) = \frac{1249}{1250}

sec(θ)= 9/7

sec (θ) = \frac{9}{7}

4cos(x-pi)=4

4 cos (x - π) = 4

Frequently Asked Questions (FAQ)

What is the general solution for sec^2(x)-7sec(x)=0 ?
The general solution for sec^2(x)-7sec(x)=0 is x=1.42744…+2pin,x=2pi-1.42744…+2pin