What is the general solution for 4sec(θ)+2sqrt(3)=sec(θ) ?

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

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4sec(θ)+2sqrt(3)=sec(θ)

4 sec (θ) + 23 = sec (θ)

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) + 23 = sec (θ)

Solve by substitution

4 sec (θ) + 23 = sec (θ)

Let:

sec (θ) = u

4 u + 23 = u

4 u + 23 = u : u = - \frac{2 3}{3}

4 u + 23 = u

Move

23

to the right side

4 u + 23 = u

Subtract

23

from both sides

4 u + 23 - 23 = u - 23

Simplify

4 u = u - 23

4 u = u - 23

Move

u

to the left side

4 u = u - 23

Subtract

u

from both sides

4 u - u = u - 23 - 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

2 cos (3 x) - 3 = 0

cos^{2} (x) - 8 cos (x) - 1 = 0

cos (x) + 2 = 1

2 cos (2 θ) = - 1

3 cot^{2} (x) = 3 cot (x)

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