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

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

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4sec(x)+2sqrt(3)=sec(x)

4 sec (x) + 23 = sec (x)

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

Degrees

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

Solution steps

4 sec (x) + 23 = sec (x)

Solve by substitution

4 sec (x) + 23 = sec (x)

Let:

sec (x) = 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 (x)

sec (x) = - \frac{2 3}{3}

sec (x) = - \frac{2 3}{3}

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

sec (x) = - \frac{2 3}{3}

General solutions for

sec (x) = - \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}

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

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

Combine all the solutions

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

sec (x) sin (x) = 2

sec (x) cot (x) = \frac{7}{6}

tan (x) cos (x) = 0

sin^{2} (x) - 3 sin (x) cos (x) + 2 cos^{2} (x) = 0

2 csc^{2} (x) - 2 csc (x) - 2 = 0

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