What is the general solution for 5sec(θ)=4sec(θ)+sqrt(2) ?

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

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5sec(θ)=4sec(θ)+sqrt(2)

5 sec (θ) = 4 sec (θ) + 2

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

Degrees

θ = 4 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

5 sec (θ) = 4 sec (θ) + 2

Solve by substitution

5 sec (θ) = 4 sec (θ) + 2

Let:

sec (θ) = u

5 u = 4 u + 2

5 u = 4 u + 2 : u = 2

5 u = 4 u + 2

Move

4 u

to the left side

5 u = 4 u + 2

Subtract

4 u

from both sides

5 u - 4 u = 4 u + 2 - 4 u

Simplify

u = 2

u = 2

Substitute back

u = sec (θ)

sec (θ) = 2

sec (θ) = 2

sec (θ) = 2 : θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

sec (θ) = 2

General solutions for

sec (θ) = 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}

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

Combine all the solutions

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

4 sec (π x) + 6 = - 2

x, y - 1 = - 4 + 3 sin (4 x - \frac{π}{4})

64 = 81 + 25 - 90 (cos (C))

sin^{2} (a - \frac{π}{8}) = \frac{2}{3}

cos^{4} (x) - 3 cos^{2} (x) = 4

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