What is the general solution for 3tan^2(x)=19 ?

The general solution for 3tan^2(x)=19 is x=1.19256…+pin,x=-1.19256…+pin

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3tan^2(x)=19

3 tan^{2} (x) = 19

x = 1.19256 \dots + πn, x = - 1.19256 \dots + πn

Degrees

x = 68.32912 \dots^{\circ} + 18 0^{\circ} n, x = - 68.32912 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 tan^{2} (x) = 19

Solve by substitution

3 tan^{2} (x) = 19

Let:

tan (x) = u

3 u^{2} = 19

3 u^{2} = 19 : u = \frac{19}{3}, u = - \frac{19}{3}

3 u^{2} = 19

Divide both sides by

3

3 u^{2} = 19

Divide both sides by

3

\frac{3 u ^{2}}{3} = \frac{19}{3}

Simplify

u^{2} = \frac{19}{3}

u^{2} = \frac{19}{3}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{19}{3}, u = - \frac{19}{3}

Substitute back

u = tan (x)

tan (x) = \frac{19}{3}, tan (x) = - \frac{19}{3}

tan (x) = \frac{19}{3}, tan (x) = - \frac{19}{3}

tan (x) = \frac{19}{3} : x = arctan (\frac{19}{3}) + πn

tan (x) = \frac{19}{3}

Apply trig inverse properties

tan (x) = \frac{19}{3}

General solutions for

tan (x) = \frac{19}{3}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{19}{3}) + πn

x = arctan (\frac{19}{3}) + πn

tan (x) = - \frac{19}{3} : x = arctan (- \frac{19}{3}) + πn

tan (x) = - \frac{19}{3}

Apply trig inverse properties

tan (x) = - \frac{19}{3}

General solutions for

tan (x) = - \frac{19}{3}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{19}{3}) + πn

x = arctan (- \frac{19}{3}) + πn

Combine all the solutions

x = arctan (\frac{19}{3}) + πn, x = arctan (- \frac{19}{3}) + πn

Show solutions in decimal form

x = 1.19256 \dots + πn, x = - 1.19256 \dots + πn

What is the general solution for 3tan^2(x)=19 ?
The general solution for 3tan^2(x)=19 is x=1.19256…+pin,x=-1.19256…+pin