What is the general solution for tan(x)sec(x)+3tan(x)=0 ?

The general solution for tan(x)sec(x)+3tan(x)=0 is x=pin,x=1.91063…+2pin,x=-1.91063…+2pin

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tan(x)sec(x)+3tan(x)=0

tan (x) sec (x) + 3 tan (x) = 0

x = πn, x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 109.47122 \dots^{\circ} + 36 0^{\circ} n, x = - 109.47122 \dots^{\circ} + 36 0^{\circ} n

Solution steps

tan (x) sec (x) + 3 tan (x) = 0

Factor

tan (x) sec (x) + 3 tan (x) : tan (x) (sec (x) + 3)

tan (x) sec (x) + 3 tan (x)

Factor out common term

tan (x)

= tan (x) (sec (x) + 3)

tan (x) (sec (x) + 3) = 0

Solving each part separately

tan (x) = 0 or sec (x) + 3 = 0

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

sec (x) + 3 = 0 : x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

sec (x) + 3 = 0

Move

3

to the right side

sec (x) + 3 = 0

Subtract

3

from both sides

sec (x) + 3 - 3 = 0 - 3

Simplify

sec (x) = - 3

sec (x) = - 3

Apply trig inverse properties

sec (x) = - 3

General solutions for

sec (x) = - 3

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

x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

Combine all the solutions

x = πn, x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

Show solutions in decimal form

x = πn, x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

\frac{csc ( b )}{csc ( b )} = \frac{1}{1 - sin ^{2} ( b )}

3 sec (4 x) = 2

2 cos (x) - 1 = cos (2 x)

0 = 4 cos (2 θ)

tan^{2} (x) cos (x) = tan^{2} (x)

What is the general solution for tan(x)sec(x)+3tan(x)=0 ?
The general solution for tan(x)sec(x)+3tan(x)=0 is x=pin,x=1.91063…+2pin,x=-1.91063…+2pin