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

The general solution for 3tan^2(x/3)-1=0 is x= pi/2+3pin,x=-pi/2+3pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

3tan^2(x/3)-1=0

Solution

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

Solution

x = \frac{π}{2} + 3 πn, x = - \frac{π}{2} + 3 πn

Degrees

x = 9 0^{\circ} + 54 0^{\circ} n, x = - 9 0^{\circ} + 54 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

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

3 u^{2} - 1 = 0

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

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 (\frac{x}{3})

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

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

tan (\frac{x}{3}) = \frac{1}{3} : x = \frac{π}{2} + 3 πn

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Solve

\frac{x}{3} = arctan (\frac{1}{3}) + πn : x = \frac{π}{2} + 3 πn

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

Simplify

arctan (\frac{1}{3}) + πn : \frac{π}{6} + πn

arctan (\frac{1}{3}) + πn

Use the following trivial identity:

arctan (\frac{1}{3}) = \frac{π}{6}

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= \frac{π}{6} + πn

\frac{x}{3} = \frac{π}{6} + πn

Multiply both sides by

3

\frac{x}{3} = \frac{π}{6} + πn

Multiply both sides by

3

\frac{3 x}{3} = 3 \cdot \frac{π}{6} + 3 πn

Simplify

\frac{3 x}{3} = 3 \cdot \frac{π}{6} + 3 πn

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

3 \cdot \frac{π}{6} + 3 πn : \frac{π}{2} + 3 πn

3 \cdot \frac{π}{6} + 3 πn

3 \cdot \frac{π}{6} = \frac{π}{2}

3 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 3}{6}

Cancel the common factor:

3

= \frac{π}{2}

= \frac{π}{2} + 3 πn

x = \frac{π}{2} + 3 πn

x = \frac{π}{2} + 3 πn

x = \frac{π}{2} + 3 πn

x = \frac{π}{2} + 3 πn

tan (\frac{x}{3}) = - \frac{1}{3} : x = - \frac{π}{2} + 3 πn

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Solve

\frac{x}{3} = arctan (- \frac{1}{3}) + πn : x = - \frac{π}{2} + 3 πn

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

Simplify

arctan (- \frac{1}{3}) + πn : - \frac{π}{6} + πn

arctan (- \frac{1}{3}) + πn

arctan (- \frac{1}{3}) = - \frac{π}{6}

arctan (- \frac{1}{3})

Use the following property:

arctan (- x) = - arctan (x)

arctan (- \frac{1}{3}) = - arctan (\frac{1}{3})

= - arctan (\frac{1}{3})

Use the following trivial identity:

arctan (\frac{1}{3}) = \frac{π}{6}

arctan (\frac{1}{3})

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= \frac{π}{6}

= - \frac{π}{6}

= - \frac{π}{6} + πn

\frac{x}{3} = - \frac{π}{6} + πn

Multiply both sides by

3

\frac{x}{3} = - \frac{π}{6} + πn

Multiply both sides by

3

\frac{3 x}{3} = - 3 \cdot \frac{π}{6} + 3 πn

Simplify

\frac{3 x}{3} = - 3 \cdot \frac{π}{6} + 3 πn

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

- 3 \cdot \frac{π}{6} + 3 πn : - \frac{π}{2} + 3 πn

- 3 \cdot \frac{π}{6} + 3 πn

3 \cdot \frac{π}{6} = \frac{π}{2}

3 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 3}{6}

Cancel the common factor:

3

= \frac{π}{2}

= - \frac{π}{2} + 3 πn

x = - \frac{π}{2} + 3 πn

x = - \frac{π}{2} + 3 πn

x = - \frac{π}{2} + 3 πn

x = - \frac{π}{2} + 3 πn

Combine all the solutions

x = \frac{π}{2} + 3 πn, x = - \frac{π}{2} + 3 πn

Popular Examples

cos(x)= 7/10

cos (x) = \frac{7}{10}

2cos(2x)+3=2

2 cos (2 x) + 3 = 2

2cos(x)=sec(x)

2 cos (x) = sec (x)

tan(5x)=-1

tan (5 x) = - 1

solvefor x,7y^2+sin(3x)=12-y^4 solve for

x, 7 y^{2} + sin (3 x) = 12 - y^{4}

Frequently Asked Questions (FAQ)

What is the general solution for 3tan^2(x/3)-1=0 ?
The general solution for 3tan^2(x/3)-1=0 is x= pi/2+3pin,x=-pi/2+3pin