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

The general solution for tan^2(x)=sqrt(3) is x=0.92103…+pin,x=-0.92103…+pin

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tan^2(x)=sqrt(3)

Solution

tan^{2} (x) = 3

Solution

x = 0.92103 \dots + πn, x = - 0.92103 \dots + πn

Degrees

x = 52.77113 \dots^{\circ} + 18 0^{\circ} n, x = - 52.77113 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (x) = 3

Solve by substitution

tan^{2} (x) = 3

Let:

tan (x) = u

u^{2} = 3

u^{2} = 3

For

x^{2} = f (a)

the solutions are

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

u = 3, u = - 3

3

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{\frac{1}{2}}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

\frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{1 \cdot 1}{2 \cdot 2}

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= 3^{\frac{1}{4}}

- 3

Simplify

3

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{\frac{1}{2}}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

\frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{1 \cdot 1}{2 \cdot 2}

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= 3^{\frac{1}{4}}

Substitute back

u = tan (x)

Apply trig inverse properties

General solutions for

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

Apply trig inverse properties

General solutions for

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

Combine all the solutions

Show solutions in decimal form

x = 0.92103 \dots + πn, x = - 0.92103 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)=sqrt(3) ?
The general solution for tan^2(x)=sqrt(3) is x=0.92103…+pin,x=-0.92103…+pin