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Popular Trigonometry >

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

Solution

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

Solution

x = - \frac{1.17809 \dots}{3} + \frac{πn}{3}, x = \frac{0.39269 \dots}{3} + \frac{πn}{3}

Degrees

x = - 22. 5^{\circ} + 6 0^{\circ} n, x = 7. 5^{\circ} + 6 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (3 x) = u

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

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

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

Multiply both sides by

1 - u^{2}

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

Multiply both sides by

1 - u^{2}

\frac{2 u}{1 - u ^{2}} (1 - u^{2}) = 1 \cdot (1 - u^{2})

Simplify

\frac{2 u}{1 - u ^{2}} (1 - u^{2}) = 1 \cdot (1 - u^{2})

Simplify

\frac{2 u}{1 - u ^{2}} (1 - u^{2}) : 2 u

\frac{2 u}{1 - u ^{2}} (1 - u^{2})

Multiply fractions:

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

= \frac{2 u ( 1 - u ^{2} )}{1 - u ^{2}}

Cancel the common factor:

1 - u^{2}

= 2 u

Simplify

1 \cdot (1 - u^{2}) : 1 - u^{2}

1 \cdot (1 - u^{2})

Multiply:

1 \cdot (1 - u^{2}) = (1 - u^{2})

= (1 - u^{2})

Remove parentheses:

(a) = a

= 1 - u^{2}

2 u = 1 - u^{2}

2 u = 1 - u^{2}

2 u = 1 - u^{2}

Solve

2 u = 1 - u^{2} : u = - 1 - 2, u = 2 - 1

2 u = 1 - u^{2}

Switch sides

1 - u^{2} = 2 u

Move

2 u

to the left side

1 - u^{2} = 2 u

Subtract

2 u

from both sides

1 - u^{2} - 2 u = 2 u - 2 u

Simplify

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

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

Write in the standard form

a x^{2} + b x + c = 0

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 1, b = - 2, c = 1

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

(- 2)^{2} - 4 (- 1) \cdot 1 = 22

(- 2)^{2} - 4 (- 1) \cdot 1

Apply rule

- (- a) = a

= (- 2)^{2} + 4 \cdot 1 \cdot 1

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 2)^{2} = 2^{2}

= 2^{2} + 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 2^{2} + 4

2^{2} = 4

= 4 + 4

Add the numbers:

4 + 4 = 8

= 8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

u_{1, 2} = \frac{- ( - 2 ) \pm 2 2}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- ( - 2 ) + 2 2}{2 ( - 1 )}, u_{2} = \frac{- ( - 2 ) - 2 2}{2 ( - 1 )}

u = \frac{- ( - 2 ) + 2 2}{2 ( - 1 )} : - 1 - 2

\frac{- ( - 2 ) + 2 2}{2 ( - 1 )}

Remove parentheses:

(- a) = - a, - (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 + 2 2}{- 2}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2 + 2 2}{2}

Cancel

\frac{2 + 2 2}{2} : 1 + 2

\frac{2 + 2 2}{2}

Factor

2 + 22 : 2 (1 + 2)

2 + 22

Rewrite as

= 2 \cdot 1 + 22

Factor out common term

2

= 2 (1 + 2)

= \frac{2 ( 1 + 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 + 2

= - (1 + 2)

Distribute parentheses

= - (1) - (2)

Apply minus-plus rules

+ (- a) = - a

= - 1 - 2

u = \frac{- ( - 2 ) - 2 2}{2 ( - 1 )} : 2 - 1

\frac{- ( - 2 ) - 2 2}{2 ( - 1 )}

Remove parentheses:

(- a) = - a, - (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 - 2 2}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

2 - 22 = - (22 - 2)

= \frac{2 2 - 2}{2}

Factor

22 - 2 : 2 (2 - 1)

22 - 2

Rewrite as

= 22 - 2 \cdot 1

Factor out common term

2

= 2 (2 - 1)

= \frac{2 ( 2 - 1 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 - 1

The solutions to the quadratic equation are:

u = - 1 - 2, u = 2 - 1

u = - 1 - 2, u = 2 - 1

Verify Solutions

Find undefined (singularity) points:

u = 1, u = - 1

Take the denominator(s) of

\frac{2 u}{1 - u ^{2}}

and compare to zero

Solve

1 - u^{2} = 0 : u = 1, u = - 1

1 - u^{2} = 0

Move

1

to the right side

1 - u^{2} = 0

Subtract

1

from both sides

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

Simplify

- u^{2} = - 1

- u^{2} = - 1

Divide both sides by

- 1

- u^{2} = - 1

Divide both sides by

- 1

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

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply radical rule:

1 = 1

= 1

- 1 = - 1

- 1

Apply radical rule:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

The following points are undefined

u = 1, u = - 1

Combine undefined points with solutions:

u = - 1 - 2, u = 2 - 1

Substitute back

u = tan (3 x)

tan (3 x) = - 1 - 2, tan (3 x) = 2 - 1

tan (3 x) = - 1 - 2, tan (3 x) = 2 - 1

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

tan (3 x) = - 1 - 2

Apply trig inverse properties

tan (3 x) = - 1 - 2

General solutions for

tan (3 x) = - 1 - 2

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

3 x = arctan (- 1 - 2) + πn

3 x = arctan (- 1 - 2) + πn

Solve

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

3 x = arctan (- 1 - 2) + πn

Simplify

arctan (- 1 - 2) + πn : - arctan (1 + 2) + πn

arctan (- 1 - 2) + πn

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 1 - 2) = - arctan (1 + 2)

= - arctan (1 + 2) + πn

3 x = - arctan (1 + 2) + πn

Divide both sides by

3

3 x = - arctan (1 + 2) + πn

Divide both sides by

3

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

Simplify

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

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

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

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

tan (3 x) = 2 - 1

Apply trig inverse properties

tan (3 x) = 2 - 1

General solutions for

tan (3 x) = 2 - 1

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

3 x = arctan (2 - 1) + πn

3 x = arctan (2 - 1) + πn

Solve

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

3 x = arctan (2 - 1) + πn

Divide both sides by

3

3 x = arctan (2 - 1) + πn

Divide both sides by

3

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

Simplify

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = - \frac{1.17809 \dots}{3} + \frac{πn}{3}, x = \frac{0.39269 \dots}{3} + \frac{πn}{3}

Graph

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