What is the general solution for 4sec(θ)-sqrt(3)=sqrt(3)+7sec(θ) ?

The general solution for 4sec(θ)-sqrt(3)=sqrt(3)+7sec(θ) is θ=(5pi)/6+2pin,θ=(7pi)/6+2pin

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4sec(θ)-sqrt(3)=sqrt(3)+7sec(θ)

Solution

4 sec (θ) - 3 = 3 + 7 sec (θ)

Solution

θ = \frac{5 π}{6} + 2 πn, θ = \frac{7 π}{6} + 2 πn

Degrees

θ = 15 0^{\circ} + 36 0^{\circ} n, θ = 21 0^{\circ} + 36 0^{\circ} n

Solution steps

4 sec (θ) - 3 = 3 + 7 sec (θ)

Solve by substitution

4 sec (θ) - 3 = 3 + 7 sec (θ)

Let:

sec (θ) = u

4 u - 3 = 3 + 7 u

4 u - 3 = 3 + 7 u : u = - \frac{2 3}{3}

4 u - 3 = 3 + 7 u

Move

3

to the right side

4 u - 3 = 3 + 7 u

Add

3

to both sides

4 u - 3 + 3 = 3 + 7 u + 3

Simplify

4 u - 3 + 3 = 3 + 7 u + 3

Simplify

4 u - 3 + 3 : 4 u

4 u - 3 + 3

Add similar elements:

- 3 + 3 = 0

= 4 u

Simplify

3 + 7 u + 3 : 7 u + 23

3 + 7 u + 3

Group like terms

= 7 u + 3 + 3

Add similar elements:

3 + 3 = 23

= 7 u + 23

4 u = 7 u + 23

4 u = 7 u + 23

4 u = 7 u + 23

Move

7 u

to the left side

4 u = 7 u + 23

Subtract

7 u

from both sides

4 u - 7 u = 7 u + 23 - 7 u

Simplify

- 3 u = 23

- 3 u = 23

Divide both sides by

- 3

- 3 u = 23

Divide both sides by

- 3

\frac{- 3 u}{- 3} = \frac{2 3}{- 3}

Simplify

u = - \frac{2 3}{3}

u = - \frac{2 3}{3}

Substitute back

u = sec (θ)

sec (θ) = - \frac{2 3}{3}

sec (θ) = - \frac{2 3}{3}

sec (θ) = - \frac{2 3}{3} : θ = \frac{5 π}{6} + 2 πn, θ = \frac{7 π}{6} + 2 πn

sec (θ) = - \frac{2 3}{3}

General solutions for

sec (θ) = - \frac{2 3}{3}

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}

θ = \frac{5 π}{6} + 2 πn, θ = \frac{7 π}{6} + 2 πn

θ = \frac{5 π}{6} + 2 πn, θ = \frac{7 π}{6} + 2 πn

Combine all the solutions

θ = \frac{5 π}{6} + 2 πn, θ = \frac{7 π}{6} + 2 πn

Popular Examples

1/(sqrt(15))=cos(x)

\frac{1}{15} = cos (x)

cos^2(θ)=3sin^2(θ)

cos^{2} (θ) = 3 sin^{2} (θ)

sinh(x)= 5/4

sinh (x) = \frac{5}{4}

3-4cos(A)=1

3 - 4 cos (A) = 1

3tan(x)-2cos(x)=0

3 tan (x) - 2 cos (x) = 0

Frequently Asked Questions (FAQ)

What is the general solution for 4sec(θ)-sqrt(3)=sqrt(3)+7sec(θ) ?
The general solution for 4sec(θ)-sqrt(3)=sqrt(3)+7sec(θ) is θ=(5pi)/6+2pin,θ=(7pi)/6+2pin