What is the general solution for 5sec(x)=2sec(x)+7 ?

The general solution for 5sec(x)=2sec(x)+7 is x=1.12788…+2pin,x=2pi-1.12788…+2pin

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5sec(x)=2sec(x)+7

5 sec (x) = 2 sec (x) + 7

x = 1.12788 \dots + 2 πn, x = 2 π - 1.12788 \dots + 2 πn

Degrees

x = 64.62306 \dots^{\circ} + 36 0^{\circ} n, x = 295.37693 \dots^{\circ} + 36 0^{\circ} n

Solution steps

5 sec (x) = 2 sec (x) + 7

Solve by substitution

5 sec (x) = 2 sec (x) + 7

Let:

sec (x) = u

5 u = 2 u + 7

5 u = 2 u + 7 : u = \frac{7}{3}

5 u = 2 u + 7

Move

2 u

to the left side

5 u = 2 u + 7

Subtract

2 u

from both sides

5 u - 2 u = 2 u + 7 - 2 u

Simplify

3 u = 7

3 u = 7

Divide both sides by

3

3 u = 7

Divide both sides by

3

\frac{3 u}{3} = \frac{7}{3}

Simplify

u = \frac{7}{3}

u = \frac{7}{3}

Substitute back

u = sec (x)

sec (x) = \frac{7}{3}

sec (x) = \frac{7}{3}

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

sec (x) = \frac{7}{3}

Apply trig inverse properties

sec (x) = \frac{7}{3}

General solutions for

sec (x) = \frac{7}{3}

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

x = arcsec (\frac{7}{3}) + 2 πn, x = 2 π - arcsec (\frac{7}{3}) + 2 πn

x = arcsec (\frac{7}{3}) + 2 πn, x = 2 π - arcsec (\frac{7}{3}) + 2 πn

Combine all the solutions

x = arcsec (\frac{7}{3}) + 2 πn, x = 2 π - arcsec (\frac{7}{3}) + 2 πn

Show solutions in decimal form

x = 1.12788 \dots + 2 πn, x = 2 π - 1.12788 \dots + 2 πn

sin (x) = (\frac{5}{9})

cos (θ) = \frac{2}{2}, sin (θ) = - \frac{2}{2}

cos (θ) = \frac{198.36}{229.4}

(cos (x) - 1) (tan (x) - 1) = 0

- cos (2 x + 3) = 0

What is the general solution for 5sec(x)=2sec(x)+7 ?
The general solution for 5sec(x)=2sec(x)+7 is x=1.12788…+2pin,x=2pi-1.12788…+2pin