What is the general solution for 2sec(x)+1=sec(x)+3 ?

The general solution for 2sec(x)+1=sec(x)+3 is x= pi/3+2pin,x=(5pi)/3+2pin

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2sec(x)+1=sec(x)+3

2 sec (x) + 1 = sec (x) + 3

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sec (x) + 1 = sec (x) + 3

Solve by substitution

2 sec (x) + 1 = sec (x) + 3

Let:

sec (x) = u

2 u + 1 = u + 3

2 u + 1 = u + 3 : u = 2

2 u + 1 = u + 3

Move

1

to the right side

2 u + 1 = u + 3

Subtract

1

from both sides

2 u + 1 - 1 = u + 3 - 1

Simplify

2 u = u + 2

2 u = u + 2

Move

u

to the left side

2 u = u + 2

Subtract

u

from both sides

2 u - u = u + 2 - u

Simplify

u = 2

u = 2

Substitute back

u = sec (x)

sec (x) = 2

sec (x) = 2

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

sec (x) = 2

General solutions for

sec (x) = 2

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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

- 7 sin (x) = 0

2 sin^{2} (x) - 1 = 0, 0 \leq x < 2 π

tan^{2} (x) + 2 tan (x) = - 1

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

What is the general solution for 2sec(x)+1=sec(x)+3 ?
The general solution for 2sec(x)+1=sec(x)+3 is x= pi/3+2pin,x=(5pi)/3+2pin