What is the general solution for 27tan^2(x)-9=0 ?

The general solution for 27tan^2(x)-9=0 is x=0.52359…+pin,x=-0.52359…+pin

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27tan^2(x)-9=0

27 tan^{2} (x) - 9 = 0

x = 0.52359 \dots + πn, x = - 0.52359 \dots + πn

Degrees

x = 3 0^{\circ} + 18 0^{\circ} n, x = - 3 0^{\circ} + 18 0^{\circ} n

Solution steps

27 tan^{2} (x) - 9 = 0

Solve by substitution

27 tan^{2} (x) - 9 = 0

Let:

tan (x) = u

27 u^{2} - 9 = 0

27 u^{2} - 9 = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

27 u^{2} - 9 = 0

Move

9

to the right side

27 u^{2} - 9 = 0

Add

9

to both sides

27 u^{2} - 9 + 9 = 0 + 9

Simplify

27 u^{2} = 9

27 u^{2} = 9

Divide both sides by

27

27 u^{2} = 9

Divide both sides by

27

\frac{27 u ^{2}}{27} = \frac{9}{27}

Simplify

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{3}, u = - \frac{1}{3}

Substitute back

u = tan (x)

tan (x) = \frac{1}{3}, tan (x) = - \frac{1}{3}

tan (x) = \frac{1}{3}, tan (x) = - \frac{1}{3}

tan (x) = \frac{1}{3} : x = arctan (\frac{1}{3}) + πn

tan (x) = \frac{1}{3}

Apply trig inverse properties

tan (x) = \frac{1}{3}

General solutions for

tan (x) = \frac{1}{3}

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

x = arctan (\frac{1}{3}) + πn

x = arctan (\frac{1}{3}) + πn

tan (x) = - \frac{1}{3} : x = arctan (- \frac{1}{3}) + πn

tan (x) = - \frac{1}{3}

Apply trig inverse properties

tan (x) = - \frac{1}{3}

General solutions for

tan (x) = - \frac{1}{3}

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

x = arctan (- \frac{1}{3}) + πn

x = arctan (- \frac{1}{3}) + πn

Combine all the solutions

x = arctan (\frac{1}{3}) + πn, x = arctan (- \frac{1}{3}) + πn

Show solutions in decimal form

x = 0.52359 \dots + πn, x = - 0.52359 \dots + πn

tan (x) = \frac{8}{17}

sin (x - 9 0^{\circ}) = cos (x)

cos (x) = \frac{67}{77}

(sin (x) + cos (x))^{2} = \frac{1 + 2 sin ( x )}{sec ( x )}

cot (\frac{π}{3}) = tan (x)

What is the general solution for 27tan^2(x)-9=0 ?
The general solution for 27tan^2(x)-9=0 is x=0.52359…+pin,x=-0.52359…+pin