What is the general solution for 4tan^2(x)-9=0 ?

The general solution for 4tan^2(x)-9=0 is x=0.98279…+pin,x=-0.98279…+pin

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4tan^2(x)-9=0

Solution

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

Solution

x = 0.98279 \dots + πn, x = - 0.98279 \dots + πn

Degrees

x = 56.30993 \dots^{\circ} + 18 0^{\circ} n, x = - 56.30993 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (x) = u

4 u^{2} - 9 = 0

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

4 u^{2} - 9 = 0

Move

9

to the right side

4 u^{2} - 9 = 0

Add

9

to both sides

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

Simplify

4 u^{2} = 9

4 u^{2} = 9

Divide both sides by

4

4 u^{2} = 9

Divide both sides by

4

\frac{4 u ^{2}}{4} = \frac{9}{4}

Simplify

u^{2} = \frac{9}{4}

u^{2} = \frac{9}{4}

For

x^{2} = f (a)

the solutions are

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

u = \frac{9}{4}, u = - \frac{9}{4}

\frac{9}{4} = \frac{3}{2}

\frac{9}{4}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{9}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= \frac{9}{2}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 3

= \frac{3}{2}

- \frac{9}{4} = - \frac{3}{2}

- \frac{9}{4}

Simplify

\frac{9}{4} : \frac{3}{2}

\frac{9}{4}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{9}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= \frac{9}{2}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 3

= \frac{3}{2}

= - \frac{3}{2}

u = \frac{3}{2}, u = - \frac{3}{2}

Substitute back

u = tan (x)

tan (x) = \frac{3}{2}, tan (x) = - \frac{3}{2}

tan (x) = \frac{3}{2}, tan (x) = - \frac{3}{2}

tan (x) = \frac{3}{2} : x = arctan (\frac{3}{2}) + πn

tan (x) = \frac{3}{2}

Apply trig inverse properties

tan (x) = \frac{3}{2}

General solutions for

tan (x) = \frac{3}{2}

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

x = arctan (\frac{3}{2}) + πn

x = arctan (\frac{3}{2}) + πn

tan (x) = - \frac{3}{2} : x = arctan (- \frac{3}{2}) + πn

tan (x) = - \frac{3}{2}

Apply trig inverse properties

tan (x) = - \frac{3}{2}

General solutions for

tan (x) = - \frac{3}{2}

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

x = arctan (- \frac{3}{2}) + πn

x = arctan (- \frac{3}{2}) + πn

Combine all the solutions

x = arctan (\frac{3}{2}) + πn, x = arctan (- \frac{3}{2}) + πn

Show solutions in decimal form

x = 0.98279 \dots + πn, x = - 0.98279 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4tan^2(x)-9=0 ?
The general solution for 4tan^2(x)-9=0 is x=0.98279…+pin,x=-0.98279…+pin