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

The general solution for sec(θ)+4=4sec(θ) is θ=0.72273…+2pin,θ=2pi-0.72273…+2pin

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sec(θ)+4=4sec(θ)

sec (θ) + 4 = 4 sec (θ)

θ = 0.72273 \dots + 2 πn, θ = 2 π - 0.72273 \dots + 2 πn

Degrees

θ = 41.40962 \dots^{\circ} + 36 0^{\circ} n, θ = 318.59037 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sec (θ) + 4 = 4 sec (θ)

Solve by substitution

sec (θ) + 4 = 4 sec (θ)

Let:

sec (θ) = u

u + 4 = 4 u

u + 4 = 4 u : u = \frac{4}{3}

u + 4 = 4 u

Move

4

to the right side

u + 4 = 4 u

Subtract

4

from both sides

u + 4 - 4 = 4 u - 4

Simplify

u = 4 u - 4

u = 4 u - 4

Move

4 u

to the left side

u = 4 u - 4

Subtract

4 u

from both sides

u - 4 u = 4 u - 4 - 4 u

Simplify

- 3 u = - 4

- 3 u = - 4

Divide both sides by

- 3

- 3 u = - 4

Divide both sides by

- 3

\frac{- 3 u}{- 3} = \frac{- 4}{- 3}

Simplify

u = \frac{4}{3}

u = \frac{4}{3}

Substitute back

u = sec (θ)

sec (θ) = \frac{4}{3}

sec (θ) = \frac{4}{3}

sec (θ) = \frac{4}{3} : θ = arcsec (\frac{4}{3}) + 2 πn, θ = 2 π - arcsec (\frac{4}{3}) + 2 πn

sec (θ) = \frac{4}{3}

Apply trig inverse properties

sec (θ) = \frac{4}{3}

General solutions for

sec (θ) = \frac{4}{3}

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

θ = arcsec (\frac{4}{3}) + 2 πn, θ = 2 π - arcsec (\frac{4}{3}) + 2 πn

θ = arcsec (\frac{4}{3}) + 2 πn, θ = 2 π - arcsec (\frac{4}{3}) + 2 πn

Combine all the solutions

θ = arcsec (\frac{4}{3}) + 2 πn, θ = 2 π - arcsec (\frac{4}{3}) + 2 πn

Show solutions in decimal form

θ = 0.72273 \dots + 2 πn, θ = 2 π - 0.72273 \dots + 2 πn

sin (2 x) - 4 \cdot cos (x) = 0

5 tan (θ) + 3 = 0

sin (θ) = - \frac{4}{7}, π < θ < \frac{3 π}{2}, cos (2 θ)

tan (θ) = - \frac{2}{5}

(tan (x) - 1) (3 tan (x) + 1) = 0

What is the general solution for sec(θ)+4=4sec(θ) ?
The general solution for sec(θ)+4=4sec(θ) is θ=0.72273…+2pin,θ=2pi-0.72273…+2pin