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

The general solution for 3sec(2x+3)=4 is x=pin-3/2+(0.72273…)/2 ,x=pi+pin-3/2-(0.72273…)/2

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3sec(2x+3)=4

Solution

3 sec (2 x + 3) = 4

Solution

x = πn - \frac{3}{2} + \frac{0.72273 \dots}{2}, x = π + πn - \frac{3}{2} - \frac{0.72273 \dots}{2}

Degrees

x = - 65.23885 \dots^{\circ} + 18 0^{\circ} n, x = 73.35151 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 sec (2 x + 3) = 4

Divide both sides by

3

3 sec (2 x + 3) = 4

Divide both sides by

3

\frac{3 sec ( 2 x + 3 )}{3} = \frac{4}{3}

Simplify

sec (2 x + 3) = \frac{4}{3}

sec (2 x + 3) = \frac{4}{3}

Apply trig inverse properties

sec (2 x + 3) = \frac{4}{3}

General solutions for

sec (2 x + 3) = \frac{4}{3}

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

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

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

Solve

2 x + 3 = arcsec (\frac{4}{3}) + 2 πn : x = πn - \frac{3}{2} + \frac{arcsec ( \frac{4}{3} )}{2}

2 x + 3 = arcsec (\frac{4}{3}) + 2 πn

Move

3

to the right side

2 x + 3 = arcsec (\frac{4}{3}) + 2 πn

Subtract

3

from both sides

2 x + 3 - 3 = arcsec (\frac{4}{3}) + 2 πn - 3

Simplify

2 x = arcsec (\frac{4}{3}) + 2 πn - 3

2 x = arcsec (\frac{4}{3}) + 2 πn - 3

Divide both sides by

2

2 x = arcsec (\frac{4}{3}) + 2 πn - 3

Divide both sides by

2

\frac{2 x}{2} = \frac{arcsec ( \frac{4}{3} )}{2} + \frac{2 πn}{2} - \frac{3}{2}

Simplify

\frac{2 x}{2} = \frac{arcsec ( \frac{4}{3} )}{2} + \frac{2 πn}{2} - \frac{3}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{arcsec ( \frac{4}{3} )}{2} + \frac{2 πn}{2} - \frac{3}{2} : πn - \frac{3}{2} + \frac{arcsec ( \frac{4}{3} )}{2}

\frac{arcsec ( \frac{4}{3} )}{2} + \frac{2 πn}{2} - \frac{3}{2}

Group like terms

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

Divide the numbers:

\frac{2}{2} = 1

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

Group like terms

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

x = πn - \frac{3}{2} + \frac{arcsec ( \frac{4}{3} )}{2}

x = πn - \frac{3}{2} + \frac{arcsec ( \frac{4}{3} )}{2}

x = πn - \frac{3}{2} + \frac{arcsec ( \frac{4}{3} )}{2}

Solve

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

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

Move

3

to the right side

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

Subtract

3

from both sides

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

Simplify

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 π}{2} - \frac{arcsec ( \frac{4}{3} )}{2} + \frac{2 πn}{2} - \frac{3}{2} : π + πn - \frac{3}{2} - \frac{arcsec ( \frac{4}{3} )}{2}

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

Group like terms

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

Divide the numbers:

\frac{2}{2} = 1

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

Group like terms

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

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

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

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

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

Show solutions in decimal form

x = πn - \frac{3}{2} + \frac{0.72273 \dots}{2}, x = π + πn - \frac{3}{2} - \frac{0.72273 \dots}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for 3sec(2x+3)=4 ?
The general solution for 3sec(2x+3)=4 is x=pin-3/2+(0.72273…)/2 ,x=pi+pin-3/2-(0.72273…)/2