What is the general solution for sqrt(3)*tan^2(x)-1=0 ?

The general solution for sqrt(3)*tan^2(x)-1=0 is x=0.64976…+pin,x=-0.64976…+pin

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sqrt(3)*tan^2(x)-1=0

Solution

3 \cdot tan^{2} (x) - 1 = 0

Solution

x = 0.64976 \dots + πn, x = - 0.64976 \dots + πn

Degrees

x = 37.22886 \dots^{\circ} + 18 0^{\circ} n, x = - 37.22886 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 tan^{2} (x) - 1 = 0

Solve by substitution

3 tan^{2} (x) - 1 = 0

Let:

tan (x) = u

3 u^{2} - 1 = 0

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

3 u^{2} - 1 = 0

Move

1

to the right side

3 u^{2} - 1 = 0

Add

1

to both sides

3 u^{2} - 1 + 1 = 0 + 1

Simplify

3 u^{2} = 1

3 u^{2} = 1

Divide both sides by

3

3 u^{2} = 1

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 u ^{2}}{3} : u^{2}

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

Cancel the common factor:

3

= u^{2}

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

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

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

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

For

x^{2} = f (a)

the solutions are

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

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

Substitute back

u = tan (x)

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

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arctan \frac{3}{3} + πn

x = arctan \frac{3}{3} + πn

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arctan - \frac{3}{3} + πn

x = arctan - \frac{3}{3} + πn

Combine all the solutions

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

Show solutions in decimal form

x = 0.64976 \dots + πn, x = - 0.64976 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)*tan^2(x)-1=0 ?
The general solution for sqrt(3)*tan^2(x)-1=0 is x=0.64976…+pin,x=-0.64976…+pin