What is the general solution for n=(2tan^3(2θ)-1)^{1/2} ?

The general solution for n=(2tan^3(2θ)-1)^{1/2} is θ=(arctan(\sqrt[3]{(n^2+1)/2}))/(2)+(pik)/2

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n=(2tan^3(2θ)-1)^{1/2}

Solution

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

Solution

Solution steps

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

Switch sides

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

Square both sides:

2 tan^{3} (2 θ) - 1 = n^{2}

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

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

Expand

((2 tan^{3} (2 θ) - 1)^{\frac{1}{2}})^{2} : 2 tan^{3} (2 θ) - 1

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

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2 tan^{3} (2 θ) - 1

2 tan^{3} (2 θ) - 1 = n^{2}

2 tan^{3} (2 θ) - 1 = n^{2}

Solve

2 tan^{3} (2 θ) - 1 = n^{2}

Move

1

to the right side

2 tan^{3} (2 θ) - 1 = n^{2}

Add

1

to both sides

2 tan^{3} (2 θ) - 1 + 1 = n^{2} + 1

Simplify

2 tan^{3} (2 θ) = n^{2} + 1

2 tan^{3} (2 θ) = n^{2} + 1

Divide both sides by

2

2 tan^{3} (2 θ) = n^{2} + 1

Divide both sides by

2

\frac{2 tan ^{3} ( 2 θ )}{2} = \frac{n ^{2}}{2} + \frac{1}{2}

Simplify

\frac{2 tan ^{3} ( 2 θ )}{2} = \frac{n ^{2}}{2} + \frac{1}{2}

Simplify

\frac{2 tan ^{3} ( 2 θ )}{2} : tan^{3} (2 θ)

\frac{2 tan ^{3} ( 2 θ )}{2}

Divide the numbers:

\frac{2}{2} = 1

= tan^{3} (2 θ)

Simplify

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

\frac{n ^{2}}{2} + \frac{1}{2}

Apply rule

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

= \frac{n ^{2} + 1}{2}

tan^{3} (2 θ) = \frac{n ^{2} + 1}{2}

tan^{3} (2 θ) = \frac{n ^{2} + 1}{2}

tan^{3} (2 θ) = \frac{n ^{2} + 1}{2}

For

x^{n} = f (a)

, n is odd, the solution is

Verify Solutions:

Check the solutions by plugging them into

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

Remove the ones that don't agree with the equation.

Plug

Square both sides:

n^{2} = n^{2}

Expand

Apply exponent rule:

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

Expand

Apply exponent rule:

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

= (\frac{n ^{2} + 1}{2})^{\frac{1}{3} \cdot 3}

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

\frac{1}{3} \cdot 3

Multiply fractions:

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

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

Cancel the common factor:

3

= 1

= \frac{n ^{2} + 1}{2}

= 2 \cdot \frac{n ^{2} + 1}{2}

Multiply fractions:

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

= \frac{( n ^{2} + 1 ) \cdot 2}{2}

Cancel the common factor:

2

= n^{2} + 1

= n^{2} + 1 - 1

1 - 1 = 0

= n^{2}

= n^{2}

n^{2} = n^{2}

n^{2} = n^{2}

Both sides are equal

True for all n

Verify Solutions:

n < 0

False

, n = 0

True

, n > 0

True

Combine domain interval with solution interval:

True for all n

Find the function intervals:

n < 0, n = 0, n > 0

Find the even roots arguments zeroes:

Solve

Factor

Rewrite

2

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

Rewrite

1

1^{3}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

Apply Difference of Cubes Formula:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

Refine

Apply exponent rule:

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

= (\frac{n ^{2} + 1}{2})^{\frac{1}{3} \cdot 2}

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

\frac{1}{3} \cdot 2

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{3}

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

Solve

Move

1

to the right side

Add

1

to both sides

Simplify

Divide both sides by

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

Divide both sides by

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

Simplify

Take both sides of the equation to the power of

3 : \frac{n ^{2} + 1}{2} = \frac{1}{2}

Expand

Apply exponent rule:

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

= (\frac{n ^{2} + 1}{2})^{\frac{1}{3} \cdot 3}

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

\frac{1}{3} \cdot 3

Multiply fractions:

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

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

Cancel the common factor:

3

= 1

= \frac{n ^{2} + 1}{2}

Expand

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

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

\frac{1}{3} \cdot 3

Multiply fractions:

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

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

Cancel the common factor:

3

= 1

= 2

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

Apply rule

1^{a} = 1

1^{3} = 1

= \frac{1}{2}

\frac{n ^{2} + 1}{2} = \frac{1}{2}

\frac{n ^{2} + 1}{2} = \frac{1}{2}

Solve

\frac{n ^{2} + 1}{2} = \frac{1}{2} : n = 0

\frac{n ^{2} + 1}{2} = \frac{1}{2}

Multiply both sides by

2

\frac{n ^{2} + 1}{2} = \frac{1}{2}

Multiply both sides by

2

\frac{2 ( n ^{2} + 1 )}{2} = \frac{1 \cdot 2}{2}

Simplify

\frac{2 ( n ^{2} + 1 )}{2} = \frac{1 \cdot 2}{2}

Simplify

\frac{2 ( n ^{2} + 1 )}{2} : n^{2} + 1

\frac{2 ( n ^{2} + 1 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= n^{2} + 1

Simplify

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

\frac{1 \cdot 2}{2}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

n^{2} + 1 = 1

n^{2} + 1 = 1

n^{2} + 1 = 1

Move

1

to the right side

n^{2} + 1 = 1

Subtract

1

from both sides

n^{2} + 1 - 1 = 1 - 1

Simplify

n^{2} = 0

n^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

n = 0

n = 0

Verify Solutions:

n = 0

True

Check the solutions by plugging them into

Remove the ones that don't agree with the equation.

Plug in

n = 0 :

True

Apply rule

0^{a} = 0

0^{2} = 0

Add the numbers:

0 + 1 = 1

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

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

Apply rule

1^{a} = 1

= 1

= 1 - 1

Subtract the numbers:

1 - 1 = 0

= 0

0 = 0

True

The solution is

n = 0

Solve

No Solution for

n \in R

Use the following exponent property

Rewrite the equation with

2^{\frac{2}{3}} u^{2} + 2^{\frac{1}{3}} u + 1 = 0

Solve

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

No Solution for

u \in R

2^{\frac{2}{3}} u^{2} + 2^{\frac{1}{3}} u + 1 = 0

Discriminant

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

2^{\frac{2}{3}} u^{2} + 2^{\frac{1}{3}} u + 1 = 0

For a quadratic equation of the form

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

the discriminant is

b^{2} - 4 a c

For

a = 2^{\frac{2}{3}}, b = 2^{\frac{1}{3}}, c = 1 : (2^{\frac{1}{3}})^{2} - 4 \cdot 2^{\frac{2}{3}} \cdot 1

(2^{\frac{1}{3}})^{2} - 4 \cdot 2^{\frac{2}{3}} \cdot 1

Expand

(2^{\frac{1}{3}})^{2} - 4 \cdot 2^{\frac{2}{3}} \cdot 1 : - 3 \cdot 2^{\frac{2}{3}}

(2^{\frac{1}{3}})^{2} - 4 \cdot 2^{\frac{2}{3}} \cdot 1

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

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

Apply exponent rule:

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

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

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

\frac{1}{3} \cdot 2

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{3}

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

4 \cdot 2^{\frac{2}{3}} \cdot 1 = 4 \cdot 2^{\frac{2}{3}}

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

Multiply the numbers:

4 \cdot 1 = 4

= 4 \cdot 2^{\frac{2}{3}}

= 2^{\frac{2}{3}} - 4 \cdot 2^{\frac{2}{3}}

Add similar elements:

2^{\frac{2}{3}} - 4 \cdot 2^{\frac{2}{3}} = - 3 \cdot 2^{\frac{2}{3}}

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

- 3 \cdot 2^{\frac{2}{3}}

Discriminant cannot be negative for

u \in R

The solution is

No Solution for u \in R

No Solution for n \in R

n = 0

Verify Solutions:

n = 0

True

Check the solutions by plugging them into

Remove the ones that don't agree with the equation.

Plug in

n = 0 :

True

Apply rule

0^{a} = 0

0^{2} = 0

Apply exponent rule:

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

= (\frac{0 + 1}{2})^{\frac{1}{3} \cdot 3}

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

\frac{1}{3} \cdot 3

Multiply fractions:

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

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

Cancel the common factor:

3

= 1

= \frac{0 + 1}{2}

Add the numbers:

0 + 1 = 1

= \frac{1}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 1 - 1

Subtract the numbers:

1 - 1 = 0

= 0

0 = 0

True

The solution is

n = 0

n = 0

The intervals are defined around the zeroes:

n < 0, n = 0, n > 0

Combine intervals with domain

n < 0, n = 0, n > 0

Check the solutions by plugging them into

Remove the ones that don't agree with the equation.

Plug

False

The solution is

n \geq 0

The solution is

Apply trig inverse properties

General solutions for

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

Solve

Divide both sides by

2

Divide both sides by

2

Simplify

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cos(x)=(sqrt(125))/(sqrt(174))

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Frequently Asked Questions (FAQ)

What is the general solution for n=(2tan^3(2θ)-1)^{1/2} ?
The general solution for n=(2tan^3(2θ)-1)^{1/2} is θ=(arctan(\sqrt[3]{(n^2+1)/2}))/(2)+(pik)/2